Brief Calculus: An Applied Approach, 8th Edition

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Brief Calculus: An Applied Approach, 8th Edition

Brief Calculus An Applied Approach RON LARSON The Pennsylvania State University The Behrend College with the assistanc

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Brief Calculus An Applied Approach

RON LARSON The Pennsylvania State University The Behrend College

with the assistance of

Eighth Edition

D AV I D C . FA LV O The Pennsylvania State University The Behrend College

HOUGHTON MIFFLIN C O M PA N Y Boston New York

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Publisher: Richard Stratton Sponsoring Editor: Cathy Cantin Senior Marketing Manager: Jennifer Jones Development Editor: Peter Galuardi Art and Design Manager: Jill Haber Cover Design Manager: Anne S. Katzeff Senior Photo Editor: Jennifer Meyer Dare Senior Composition Buyer: Chuck Dutton Senior New Title Project Manager: Pat O’Neill Editorial Associate: Jeannine Lawless Marketing Associate: Mary Legere Editorial Assistant: Jill Clark

Cover photo © Torsten Andreas Hoffmann/Getty Images

Copyright © 2009 by Houghton Mifflin Company. All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without the prior written permission of Houghton Mifflin Company unless such copying is expressly permitted by federal copyright law. Address inquiries to College Permissions, Houghton Mifflin Company, 222 Berkeley Street, Boston, MA 02116-3764. Printed in the U.S.A. Library of Congress Control Number: 2007925316 Instructor’s examination copy ISBN-10: 0-547-00480-X ISBN-13: 978-0-547-00480-8 For orders, use student text ISBNs ISBN-10: 0-618-95847-9 ISBN-13: 978-0-618-95847-4 123456789–DOW– 11 10 09 08 07

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

iii

Contents

Contents A Word from the Author (Preface) A Plan for You as a Student ix Features xiii

0

A Precalculus Review 0.1 0.2 0.3 0.4 0.5

1

vii

1

The Real Number Line and Order 2 Absolute Value and Distance on the Real Number Line Exponents and Radicals 13 Factoring Polynomials 19 Fractions and Rationalization 25

Functions, Graphs, and Limits

33

1.1 The Cartesian Plane and the Distance Formula 1.2 Graphs of Equations 43 1.3 Lines in the Plane and Slope 56 Mid-Chapter Quiz 68 1.4 Functions 69 1.5 Limits 82 1.6 Continuity 94 Chapter 1 Algebra Review 105 Chapter Summary and Study Strategies 107 Review Exercises 109 Chapter Test 113

2

34

Differentiation 2.1 The Derivative and the Slope of a Graph 2.2 Some Rules for Differentiation 126 2.3 Rates of Change: Velocity and Marginals 2.4 The Product and Quotient Rules 153 Mid-Chapter Quiz 164 2.5 The Chain Rule 165 2.6 Higher-Order Derivatives 174 2.7 Implicit Differentiation 181 2.8 Related Rates 188 Chapter 2 Algebra Review 196 Chapter Summary and Study Strategies 198 Review Exercises 200 Chapter Test 204

8

114 115 138

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

iv

Contents

3

Applications of the Derivative

205

3.1 Increasing and Decreasing Functions 206 3.2 Extrema and the First-Derivative Test 215 3.3 Concavity and the Second-Derivative Test 225 3.4 Optimization Problems 235 Mid-Chapter Quiz 244 3.5 Business and Economics Applications 245 3.6 Asymptotes 255 3.7 Curve Sketching: A Summary 266 3.8 Differentials and Marginal Analysis 275 Chapter 3 Algebra Review 283 Chapter Summary and Study Strategies 285 Review Exercises 287 Chapter Test 291

4

Exponential and Logarithmic Functions

292

4.1 Exponential Functions 293 4.2 Natural Exponential Functions 299 4.3 Derivatives of Exponential Functions 308 Mid-Chapter Quiz 316 4.4 Logarithmic Functions 317 4.5 Derivatives of Logarithmic Functions 326 4.6 Exponential Growth and Decay 335 Chapter 4 Algebra Review 344 Chapter Summary and Study Strategies 346 Review Exercises 348 Chapter Test 352

5

Integration and Its Applications

353

Antiderivatives and Indefinite Integrals 354 Integration by Substitution and the General Power Rule 365 5.3 Exponential and Logarithmic Integrals 374 Mid-Chapter Quiz 381 5.4 Area and the Fundamental Theorem of Calculus 382 5.5 The Area of a Region Bounded by Two Graphs 394 5.6 The Definite Integral as the Limit of a Sum 403 Chapter 5 Algebra Review 409 Chapter Summary and Study Strategies 411 Review Exercises 413 Chapter Test 417 5.1 5.2

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

v

Contents

6

Techniques of Integration

418

6.1 Integration by Parts and Present Value 419 6.2 Partial Fractions and Logistic Growth 429 6.3 Integration Tables 439 Mid-Chapter Quiz 449 6.4 Numerical Integration 450 6.5 Improper Integrals 459 Chapter 6 Algebra Review 470 Chapter Summary and Study Strategies 472 Review Exercises 474 Chapter Test 477

7

Functions of Several Variables

478

7.1 The Three-Dimensional Coordinate System 479 7.2 Surfaces in Space 487 7.3 Functions of Several Variables 496 7.4 Partial Derivatives 505 7.5 Extrema of Functions of Two Variables 516 Mid-Chapter Quiz 525 7.6 Lagrange Multipliers 526 7.7 Least Squares Regression Analysis 535 7.8 Double Integrals and Area in the Plane 545 7.9 Applications of Double Integrals 553 Chapter 7 Algebra Review 561 Chapter Summary and Study Strategies 563 Review Exercises 565 Chapter Test 569

Appendices Appendix A: Appendix B: B.1 B.2

Alternative Introduction to the Fundamental Theorem of Calculus Formulas A10

A1

Differentiation and Integration Formulas A10 Formulas from Business and Finance A14

Appendix C: Differential Equations (web only)* C.1 C.2 C.3 C.4

Solutions of Differential Equations Separation of Variables First-Order Linear Differential Equations Applications of Differential Equations

Appendix D: Properties and Measurement (web only)* D.1 D.2

Review of Algebra, Geometry, and Trigonometry Units of Measurements

Appendix E: Graphing Utility Programs (web only)* E.1

Graphing Utility Programs

Answers to Selected Exercises Answers to Checkpoints A91 Index A103

A17

*Available at the text-specific website at college.hmco.com/pic/larsonBrief8e

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

A Word from the Author

vii

A Word from the Author Welcome to Brief Calculus: An Applied Approach, Eighth Edition. In this revision, I focused not only on providing a meaningful revision to the text, but also a completely integrated learning program. Applied calculus students are a diverse group with varied interests and backgrounds. The revision strives to address the diversity and the different learning styles of students. I also aimed to alleviate and remove obstacles that prevent students from mastering the material.

An Enhanced Text The table of contents was streamlined to enable instructors to spend more time on each topic. This added time will give students a better understanding of the concepts and help them to master the material. Real data and applications were updated, rewritten, and added to address more modern topics, and data was gathered from news sources, current events, industry, world events, and government. Exercises derived from other disciplines’ textbooks are included to show the relevance of the calculus to students’ majors. I hope these changes will give students a clear picture that the math they are learning exists beyond the classroom. Two new chapter tests were added: a Mid-Chapter Quiz and a Chapter Test. The Mid-Chapter quiz gives students the opportunity to discover any topics they might need to study further before they progress too far into the chapter. The Chapter Test allows students to identify and strengthen any weaknesses in advance of an exam. Several new section-level features were added to promote further mastery of the concepts. ■





Concept Checks appear at the end of each section, immediately before the exercise sets. They ask non-computational questions designed to test students’ basic understanding of that sections’ concepts. Make a Decision exercises and examples ask open-ended questions that force students to apply concepts to real-world situations. Extended Applications are more in-depth, applied exercises requiring students to work with large data sets and often involve work in creating or analyzing models.

I hope the combination of these new features with the existing features will promote a deeper understanding of the mathematics.

Enhanced Resources Although the textbook often forms the basis of the course, today’s students often find greater value in an integrated text and technology program. With that in mind, I worked with the publisher to enhance the online and media resources available to students, to provide them with a complete learning program.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

viii

A Word from the Author

An HM MathSPACE course has been developed with dynamic, algorithmic exercises tied to exercises within the text. These exercises provide students with unlimited practice for complete mastery of the topics. An additional resource for the 8th edition is a Multimedia Online eBook. This eBook breaks the physical constraints of a traditional text and binds a number of multimedia assets and features to the text itself. Based in Flash, students can read the text, watch the videos when they need extra explanation, view enlarged math graphs, and more. The eBook promotes multiple learning styles and provides students with an engaging learning experience. For students who work best in groups or whose schedules don’t allow them to come to office hours, Calc Chat is now available with this edition. Calc Chat (located at www.CalcChat.com) provides solutions to exercises. Calc Chat also has a moderated online forum for students to discuss any issues they may be having with their calculus work. I hope you enjoy the enhancements made to the eighth edition. I believe the whole suite of learning options available to students will enable any student to master applied calculus.

Ron Larson

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

A Plan for You as a Student

ix

A Plan for You as a Student Study Strategies Your success in mathematics depends on your active participation both in class and outside of class. Because the material you learn each day builds on the material you have learned previously, it is important that you keep up with your course work every day and develop a clear plan of study. This set of guidelines highlights key study strategies to help you learn how to study mathematics. Preparing for Class The syllabus your instructor provides is an invaluable resource that outlines the major topics to be covered in the course. Use it to help you prepare. As a general rule, you should set aside two to four hours of study time for each hour spent in class. Being prepared is the first step toward success. Before class: ■ Review your notes from the previous class. ■ Read the portion of the text that will be covered in class. Keeping Up Another important step toward success in mathematics involves your ability to keep up with the work. It is very easy to fall behind, especially if you miss a class. To keep up with the course work, be sure to: ■ Attend every class. Bring your text, a notebook, a pen or pencil, and a calculator (scientific or graphing). If you miss a class, get the notes from a classmate as soon as possible and review them carefully. ■ Participate in class. As mentioned above, if there is a topic you do not understand, ask about it before the instructor moves on to a new topic. ■ Take notes in class. After class, read through your notes and add explanations so that your notes make sense to you. Fill in any gaps and note any questions you might have. Getting Extra Help It can be very frustrating when you do not understand concepts and are unable to complete homework assignments. However, there are many resources available to help you with your studies. ■ Your instructor may have office hours. If you are feeling overwhelmed and need help, make an appointment to discuss your difficulties with your instructor. ■ Find a study partner or a study group. Sometimes it helps to work through problems with another person. ■ Special assistance with algebra appears in the Algebra Reviews, which appear throughout each chapter. These short reviews are tied together in the larger Algebra Review section at the end of each chapter. Preparing for an Exam The last step toward success in mathematics lies in how you prepare for and complete exams. If you have followed the suggestions given above, then you are almost ready for exams. Do not assume that you can cram for the exam the night before—this seldom works. As a final preparation for the exam: ■ When you study for an exam, first look at all definitions, properties, and formulas until you know them. Review your notes and the portion of the text that will be covered on the exam. Then work as many exercises as you can, especially any kinds of exercises that have given you trouble in the past, reworking homework problems as necessary. ■ Start studying for your exam well in advance (at least a week). The first day or two, study only about two hours. Gradually increase your study time each day. Be completely prepared for the exam two days in advance. Spend the final day just building confidence so you can be relaxed during the exam. For a more comprehensive list of study strategies, please visit college.hmco.com/pic/larsonBrief8e.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

x

Supplements

Get more value from your textbook! Supplements for the Instructor

Supplements for the Student

Digital Instructor’s Solution Manual Found on the instructor website, this manual contains the complete, worked-out solutions for all the exercises in the text.

Student Solutions Guide This guide contains complete solutions to all odd-numbered exercises in the text. Excel Made Easy CD This CD uses easy-to-follow videos to help students master mathematical concepts introduced in class. Electronic spreadsheets and detailed tutorials are included.

Instructor and Student Websites The Instructor and Student websites at college.hmco.com/pic/larsonBrief8e contain an abundance of resources for teaching and learning, such as Note Taking Guides, a Graphing Calculator Guide, Digital Lessons, ACE Practice Tests, and a graphing calculator simulator. Instruction DVDs Hosted by Dana Mosely and captioned for the hearing-impaired, these DVDs cover all sections in the text. Ideal for promoting individual study and review, these comprehensive DVDs also support students in online courses or those who have missed a lecture. HM MathSPACE® HM MathSPACE encompasses the interactive online products and services integrated with Houghton Mifflin mathematics programs. Students and instructors can access HM MathSPACE content through text-specific Student and Instructor websites and via online learning platforms including WebAssign as well as Blackboard®, WebCT®, and other course management systems. HM Testing™ (powered by Diploma™) HM Testing (powered by Diploma) provides instructors with a wide array of algorithmic items along with improved functionality and ease of use. HM Testing offers all the tools needed to create, deliver, and customize multiple types of tests—including authoring and editing algorithmic questions. In addition to producing an unlimited number of tests for each chapter, including cumulative tests and final exams, HM Testing also offers instructors the ability to deliver tests online, or by paper and pencil. Online Course Content for Blackboard®, WebCT®, and eCollege® Deliver program or text-specific Houghton Mifflin content online using your institution’s local course management system. Houghton Mifflin offers homework, tutorials, videos, and other resources formatted for Blackboard®, WebCT®, eCollege®, and other course management systems. Add to an existing online course or create a new one by selecting from a wide range of powerful learning and instructional materials.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Acknowledgments

xi

Acknowledgments I would like to thank the many people who have helped me at various stages of this project during the past 27 years. Their encouragement, criticisms, and suggestions have been invaluable. Thank you to all of the instructors who took the time to review the changes to this edition and provide suggestions for improving it. Without your help this book would not be possible.

Reviewers of the Eighth Edition Lateef Adelani, Harris-Stowe State University, Saint Louis; Frederick Adkins, Indiana University of Pennsylvania; Polly Amstutz, University of Nebraska at Kearney; Judy Barclay, Cuesta College; Jean Michelle Benedict, Augusta State University; Ben Brink, Wharton County Junior College; Jimmy Chang, St. Petersburg College; Derron Coles, Oregon State University; David French, Tidewater Community College; Randy Gallaher, Lewis & Clark Community College; Perry Gillespie, Fayetteville State University; Walter J. Gleason, Bridgewater State College; Larry Hoehn, Austin Peay State University; Raja Khoury, Collin County Community College; Ivan Loy, Front Range Community College; Lewis D. Ludwig, Denison University; Augustine Maison, Eastern Kentucky University; John Nardo, Oglethorpe University; Darla Ottman, Elizabethtown Community & Technical College; William Parzynski, Montclair State University; Laurie Poe, Santa Clara University; Adelaida Quesada, Miami Dade College—Kendall; Brooke P. Quinlan, Hillsborough Community College; David Ray, University of Tennessee at Martin; Carol Rychly, Augusta State University; Mike Shirazi, Germanna Community College; Rick Simon, University of La Verne; Marvin Stick, University of Massachusetts—Lowell; Devki Talwar, Indiana University of Pennsylvania; Linda Taylor, Northern Virginia Community College; Stephen Tillman, Wilkes University; Jay Wiestling, Palomar College; John Williams, St. Petersburg College; Ted Williamson, Montclair State University

Reviewers of the Seventh Edition George Anastassiou, University of Memphis; Keng Deng, University of Louisiana at Lafayette; Jose Gimenez, Temple University; Shane Goodwin, Brigham Young University of Idaho; Harvey Greenwald, California Polytechnic State University; Bernadette Kocyba, J. Sergeant Reynolds Community College; Peggy Luczak, Camden County College; Randall McNiece, San Jacinto College; Scott Perkins, Lake Sumter Community College

Reviewers of Previous Editions Carol Achs, Mesa Community College; David Bregenzer, Utah State University; Mary Chabot, Mt. San Antonio College; Joseph Chance, University of Texas—Pan American; John Chuchel, University of California; Miriam E. Connellan, Marquette University; William Conway, University of Arizona; Karabi Datta, Northern Illinois University; Roger A. Engle, Clarion University of Pennsylvania; Betty Givan, Eastern Kentucky University; Mark Greenhalgh,

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

xii

Acknowledgments

Fullerton College; Karen Hay, Mesa Community College; Raymond Heitmann, University of Texas at Austin; William C. Huffman, Loyola University of Chicago; Arlene Jesky, Rose State College; Ronnie Khuri, University of Florida; Duane Kouba, University of California—Davis; James A. Kurre, The Pennsylvania State University; Melvin Lax, California State University—Long Beach; Norbert Lerner, State University of New York at Cortland; Yuhlong Lio, University of South Dakota; Peter J. Livorsi, Oakton Community College; Samuel A. Lynch, Southwest Missouri State University; Kevin McDonald, Mt. San Antonio College; Earl H. McKinney, Ball State University; Philip R. Montgomery, University of Kansas; Mike Nasab, Long Beach City College; Karla Neal, Louisiana State University; James Osterburg, University of Cincinnati; Rita Richards, Scottsdale Community College; Stephen B. Rodi, Austin Community College; Yvonne Sandoval-Brown, Pima Community College; Richard Semmler, Northern Virginia Community College—Annandale; Bernard Shapiro, University of Massachusetts—Lowell; Jane Y. Smith, University of Florida; DeWitt L. Sumners, Florida State University; Jonathan Wilkin, Northern Virginia Community College; Carol G. Williams, Pepperdine University; Melvin R. Woodard, Indiana University of Pennsylvania; Carlton Woods, Auburn University at Montgomery; Jan E. Wynn, Brigham Young University; Robert A. Yawin, Springfield Technical Community College; Charles W. Zimmerman, Robert Morris College My thanks to David Falvo, The Behrend College, The Pennsylvania State University, for his contributions to this project. My thanks also to Robert Hostetler, The Behrend College, The Pennsylvania State University, and Bruce Edwards, University of Florida, for their significant contributions to previous editions of this text. I would also like to thank the staff at Larson Texts, Inc. who assisted with proofreading the manuscript, preparing and proofreading the art package, and checking and typesetting the supplements. On a personal level, I am grateful to my spouse, Deanna Gilbert Larson, for her love, patience, and support. Also, a special thanks goes to R. Scott O’Neil. If you have suggestions for improving this text, please feel free to write to me. Over the past two decades I have received many useful comments from both instructors and students, and I value these comments very highly.

Ron Larson

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

xiii

Features

How to get the most out of your textbook . . . CHAPTER OPENERS

2

Differentiation

© Schlegelmilch/Corbis

Each opener has an applied example of a core topic from the chapter. The section outline provides a comprehensive overview of the material being presented.

2.1 2.2 2.3

2.4 2.5 2.6 2.7 2.8

The Derivative and the Slope of a Graph Some Rules for Differentiation Rates of Change: Velocity and Marginals The Product and Quotient Rules The Chain Rule Higher-Order Derivatives Implicit Differentiation Related Rates

Higher-order derivatives are used to determine the acceleration function of a sports car. The acceleration function shows the changes in the car’s velocity. As the car reaches its “cruising”speed, is the acceleration increasing or decreasing? (See Section 2.6, Exercise 45.)

Applications Differentiation has many real-life applications. The applications listed below represent a sample of the applications in this chapter. ■ ■ ■ ■ ■

Sales, Exercise 61, page 137 Political Fundraiser, Exercise 63, page 137 Make a Decision: Inventory Replenishment, Exercise 65, page 163 Modeling Data, Exercise 51, page 180 Health: U.S. HIV/AIDS Epidemic, Exercise 47, page 187

114

SECTION 2.1

The Derivative and the Slope of a Graph

115

Section 2.1 ■ Identify tangent lines to a graph at a point.

The Derivative and the Slope of a Graph

■ Approximate the slopes of tangent lines to graphs at points. ■ Use the limit definition to find the slopes of graphs at points. ■ Use the limit definition to find the derivatives of functions. ■ Describe the relationship between differentiability and continuity.

SECTION OBJECTIVES A bulleted list of learning objectives allows you the opportunity to preview what will be presented in the upcoming section.

Tangent Line to a Graph y

(x3, y3) (x2, y2)

(x4, y4) x

(x1, y1)

F I G U R E 2 . 1 The slope of a nonlinear graph changes from one point to another.

Calculus is a branch of mathematics that studies rates of change of functions. In this course, you will learn that rates of change have many applications in real life. In Section 1.3, you learned how the slope of a line indicates the rate at which the line rises or falls. For a line, this rate (or slope) is the same at every point on the line. For graphs other than lines, the rate at which the graph rises or falls changes from point to point. For instance, in Figure 2.1, the parabola is rising more quickly at the point 共x1, y1兲 than it is at the point 共x2, y2 兲. At the vertex 共x3, y3兲, the graph levels off, and at the point 共x4, y4兲, the graph is falling. To determine the rate at which a graph rises or falls at a single point, you can find the slope of the tangent line at the point. In simple terms, the tangent line to the graph of a function f at a point P共x1, y1兲 is the line that best approximates the graph at that point, as shown in Figure 2.1. Figure 2.2 shows other examples of tangent lines.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

xiv

Features

NEW!

45. MAKE A DECISION: FUEL COST A car is driven 15,000 miles a year and gets x miles per gallon. Assume that the average fuel cost is $2.95 per gallon. Find the annual cost of fuel C as a function of x and use this function to complete the table.

MAKE A DECISION

Multi-step exercises reinforce your problem-solving skills and mastery of concepts, as well as taking a real-life application further by testing what you know about a given problem to make a decision within the context of the problem.

x

10

15

20

25

30

35

40

C dC兾dx g

Who would benefit more from a 1 mile per gallon increase 61. MAKE A DECISION: NEGOTIATING A PRICE You in fuel efficiency—the driver who gets 15 miles per gallon decide to form a partnership with another business. Your or the driver who gets 35 miles per gallon? Explain. business determines that the demand x for your product is inversely proportional to the square of the price for x ≥ 5. (a) The price is $1000 and the demand is 16 units. Find the demand function. (b) Your partner determines that the product costs $250 per unit and the fixed cost is $10,000. Find the cost function. (c) Find the profit function and use a graphing utility to graph it. From the graph, what price would you negotiate with your partner for this product? Explain your reasoning.

CONCEPT CHECK 1. What is the name of the line that best approximates the slope of a graph at a point? 2. What is the name of a line through the point of tangency and a second point on the graph? 3. Sketch a graph of a function whose derivative is always negative. 4. Sketch a graph of a function whose derivative is always positive.

NEW!

CONCEPT CHECK

These non-computational questions appear at the end of each section and are designed to check your understanding of the concepts covered in that section.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Features

The Sum and Difference Rules

DEFINITIONS AND THEOREMS

The derivative of the sum or difference of two differentiable functions is the sum or difference of their derivatives.

All definitions and theorems are highlighted for emphasis and easy recognition.

d 关 f 共x) ⫹ g共x兲兴 ⫽ f⬘共x兲 ⫹ g⬘共x兲 dx

Sum Rule

d 关 f 共x兲 ⫺ g共x兲兴 ⫽ f⬘共x兲 ⫺ g⬘共x兲 dx

Difference Rule

Definition of Average Rate of Change

If y ⫽ f 共x兲, then the average rate of change of y with respect to x on the interval 关a, b兴 is Average rate of change ⫽ ⫽

⌬y . ⌬x

Note that f 共a兲 is the value of the function at the left endpoint of the interval, f 共b兲 is the value of the function at the right endpoint of the interval, and b ⫺ a is the width of the interval, as shown in Figure 2.18.

x

1

y g(x) = − 2 x 4 + 3x 3 − 2x

Example 9

60

1 g共x兲 ⫽ ⫺ x 4 ⫹ 3x 3 ⫺ 2x 2

40 30

at the point 共⫺1, ⫺ 2 兲. 3

20

Slope = 9 −3 −2

− 10 − 20

Using the Sum and Difference Rules

Find an equation of the tangent line to the graph of

50

SOLUTION 1

2

3

(−1, − )

4

5

7

3 2

that the slope of the graph at the point 共⫺1, ⫺ 2 兲 is 3

as shown in Figure 2.16. Using the point-slope form, you can write the equation of the tangent line at 共⫺1, ⫺ 32 兲 as shown.

✓CHECKPOINT 9

冢 32冣 ⫽ 9关x ⫺ 共⫺1兲兴

y⫺ ⫺

an equation of the C H A P T E R 2 Find Differentiation

tangent line to the graph of f 共x兲 ⫽ ⫺x2 ⫹ 3x ⫺ 2 at the Application point 共2, 0兲. ■

Example 10

x

The derivative of g共x兲 is g⬘共x兲 ⫽ ⫺2x3 ⫹ 9x2 ⫺ 2, which implies

Slope ⫽ g⬘共⫺1兲 ⫽ ⫺2共⫺1兲3 ⫹ 9共⫺1兲2 ⫺ 2 ⫽2⫹9⫺2 ⫽9

FIGURE 2.16

134

f 共b兲 ⫺ f 共a兲 b⫺a

y ⫽ 9x ⫹

Point-slope form

15 2

Equation of tangent line

Modeling Revenue

From 2000 through 2005, the revenue R (in millions of dollars per year) for Microsoft Corporation can be modeled by R ⫽ ⫺110.194t 3 ⫹ 993.98t2 ⫹ 1155.6t ⫹ 23,036, Microsoft Revenue

One way to answer this question is to find the derivative of the revenue model with respect to time.

Revenue (in millions of dollars)

R

SOLUTION

45,000 40,000 35,000 30,000 25,000 20,000 15,000 10,000 5,000

dR ⫽ ⫺330.582t 2 ⫹ 1987.96t ⫹ 1155.6, 0 ≤ t ≤ 5 dt In 2001 (when t ⫽ 1), the rate of change of the revenue with respect to time is given by

Slope ≈ 2813

⫺330.582共1兲2 ⫹ 1987.96共1兲 ⫹ 1155.6 ⬇ 2813. 1

2

3

4

Year (0 ↔ 2000)

FIGURE 2.17

0 ≤ t ≤ 5

where t represents the year, with t ⫽ 0 corresponding to 2000. At what rate was Microsoft’s revenue changing in 2001? (Source: Microsoft Corporation)

5

t

Because R is measured in millions of dollars and t is measured in years, it follows that the derivative dR兾dt is measured in millions of dollars per year. So, at the end of 2001, Microsoft’s revenue was increasing at a rate of about $2813 million per year, as shown in Figure 2.17.

✓CHECKPOINT 10 From 1998 through 2005, the revenue per share R (in dollars) for McDonald’s Corporation can be modeled by R ⫽ 0.0598t 2 ⫺ 0.379t ⫹ 8.44, 8 ≤ t ≤ 15

EXAMPLES There are a wide variety of relevant examples in the text, each titled for easy reference. Many of the solutions are presented graphically, analytically, and/or numerically to provide further insight into mathematical concepts. Examples using a real-life situation are identified with the symbol.

CHECKPOINT After each example, a similar problem is presented to allow for immediate practice, and to further reinforce your understanding of the concepts just learned.

where t represents the year, with t ⫽ 8 corresponding to 1998. At what rate was McDonald’s revenue per share changing in 2003? (Source: McDonald’s Corporation) ■

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

xv

xvi

Features g

D I S C O V E RY

g

D I S C O V E RY

These projects appear before selected topics and allow you to explore concepts on your own. These boxed features are optional, so they can be omitted with no loss of continuity in the coverage of material.

Use a graphing utility to graph f 共x兲 ⫽ 2x 3 ⫺ 4x 2 ⫹ 3x ⫺ 5. On the same screen, sketch the graphs of y ⫽ x ⫺ 5, y ⫽ 2x ⫺ 5, and y ⫽ 3x ⫺ 5. Which of these lines, if any, appears to be tangent to the graph of f at the point 共0, ⫺5兲? Explain your reasoning.

146

CHAPTER 2

Differentiation

TECHNOLOGY Modeling a Demand Function

To model a demand function, you need data that indicate how many units of a product will sell at a given price. As you might imagine, such data are not easy to obtain for a new product. After a product has been on the market awhile, however, its sales history can provide the necessary data. As an example, consider the two bar graphs shown below. From these graphs, you can see that from 2001 through 2005, the number of prerecorded DVDs sold increased from about 300 million to about 1100 million. During that time, the price per unit dropped from an average price of about $18 to an average price of about $15. (Source: Kagan Research, LLC) Prerecorded DVDs

Prerecorded DVDs

p

Average price per unit (in dollars)

Number of units sold (in millions)

x

1200 1000 800 600 400 200 1

2

3

4

5

t

1

The Derivative and the Slope of a Graph

t

1

2

3

4

5

x

291.5

507.5

713.0

976.6

1072.4

Find the derivative of y with respect to t for the function

p

18.40

17.11

15.83

15.51

14.94

y ⫺ y1 ⫽ m共t ⫺ t1兲

2 y⫽ . t SOLUTION

⫺2 ⌬t ⌬t→0 t共⌬t兲共t ⫹ ⌬t兲

y ⫺ 2 ⫽ ⫺2共t ⫺ 1兲 or

⫽ lim

y ⫽ ⫺2t ⫹ 4.

⫽ lim

By graphing y ⫽ 2兾t and y ⫽ ⫺2t ⫹ 4 in the same viewing window, as shown below, you can confirm that the line is tangent to the graph at the point 共1, 2兲.*

⌬t→0

5

t

Set up difference quotient.

By entering the ordered pairs 共x, p兲 into a graphing utility, you can find that the power model for the demand for prerecorded DVDs is: p ⫽ 44.55x⫺0.155, 291.5 ≤ x ≤ 1072.4. A graph of this demand function and its data points is shown below 20

Use f 共t兲 ⫽ 2兾t.

Expand terms. 200

5

1100

Factor and divide out. Simplify. Evaluate the limit.

So, the derivative of y with respect to t is dy 2 ⫽ ⫺ 2. dt t

6

Remember that the derivative of a function gives you a formula for finding the slope of the tangent line at any point on the graph of the function. For example, the slope of the tangent line to the graph of f at the point 共1, 2兲 is given by f⬘ 共1兲 ⫽ ⫺

−4

⫺2 t共t ⫹ ⌬t兲

2 ⫽⫺2 t

4

−6

Consider y ⫽ f 共t兲, and use the limit process as shown.

dy f 共t ⫹ ⌬t兲 ⫺ f 共t兲 ⫽ lim ⌬t→0 dt ⌬t 2 2 ⫺ t ⫹ ⌬t t ⫽ lim ⌬t→0 ⌬t 2t ⫺ 2t ⫺ 2 ⌬t t共t ⫹ ⌬t兲 ⫽ lim ⌬t→0 ⌬t

4

The information in the two bar graphs is combined in the table, where x represents the units sold (in millions) and p represents the price (in dollars).

Finding a Derivative

Example 7

You can use a graphing utility to confirm the result given in Example 7. One way to do this is to choose a point on the graph of y ⫽ 2兾t, such as 共1, 2兲, and find the equation of the tangent line at that point. Using the derivative found in the example, you know that the slope of the tangent line when t ⫽ 1 is m ⫽ ⫺2. This means that the tangent line at the point 共1, 2兲 is

3

121

In many applications, it is convenient to use a variable other than x as the independent variable. Example 7 shows a function that uses t as the independent variable. TECHNOLOGY

2

Year (1 ↔ 2001)

Year (1 ↔ 2001)

SECTION 2.1

20 18 16 14 12 10 8 6 4 2

2 ⫽ ⫺2. 12

To find the slopes of the graph at other points, substitute the t-coordinate of the point into the derivative, as shown below. Point

t-Coordinate

Slope

✓CHECKPOINT 7

共2, 1兲

t⫽2

m ⫽ f⬘ 共2兲 ⫽ ⫺

Find the derivative of y with respect to t for the function y ⫽ 4兾t. ■

共⫺2, ⫺1兲

t ⫽ ⫺2

2 1 ⫽⫺ 22 2 2 1 m ⫽ f⬘ 共⫺2兲 ⫽ ⫺ ⫽⫺ 共⫺2兲2 2

TECHNOLOGY BOXES These boxes appear throughout the text and provide guidance on using technology to ease lengthy calculations, present a graphical solution, or discuss where using technology can lead to misleading or wrong solutions.

*Specific calculator keystroke instructions for operations in this and other technology boxes can be found at college.hmco.com/info/larsonapplied.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Features

TECHNOLOGY EXERCISES Many exercises in the text can be solved with or without technology. The symbol identifies exercises for which students are specifically instructed to use a graphing calculator or a computer algebra system to solve the problem. Additionally, the symbol denotes exercises best solved by using a spreadsheet.

78. Credit Card Rate The average annual rate r (in percent form) for commercial bank credit cards from 2000 through 2005 can be modeled by r ⫽ 冪⫺1.7409t4 ⫹ 18.070t3 ⫺ 52.68t2 ⫹ 10.9t ⫹ 249 where t represents the year, with t ⫽ 0 corresponding to 2000. (Source: Federal Reserve Bulletin) (a) Find the derivative of this model. Which differentiation rule(s) did you use? Numerical, and Analytic (b) Use aGraphical, graphing utility to graph the derivative on theAnalysis In Exercises 63–66, use a graphing utility to graph f on interval 0 ≤ t ≤ 5. the interval [ⴚ2, 2]. Complete the table by graphically (c) Use the trace feature toslopes find theof years thegiven points. estimating the theduring graphwhich at the finance rate was changing theslopes most. analytically and compare Then evaluate the your results those obtained (d) Use the trace featurewith to find the years duringgraphically. which the finance rate was changing the least. 3 1 0 12 1 32 2 x ⫺2 ⫺1 ⫺2 ⫺2 f 共x 兲 f⬘ 共x兲 63. f 共x兲 ⫽ 14x 3 64. f 共x兲 ⫽ 12x 2 ( ) 1 3 65. f 共x兲 ⫽ ⫺ 2x 66. f 共x兲 ⫽ ⫺ 32x 2 57. Income Distribution Using the Lorenz curve in Exercise 56 and a spreadsheet, complete the table, which lists the percent of total income earned by each quintile in the United States in 2005. Quintile

Lowest

2nd

3rd

4th

Highest

Percent

Business Capsule

BUSINESS CAPSULES

AP/Wide World Photos

n 1978 Ben Cohen and Jerry Greenfield used their combined life savings of $8000 to convert an abandoned gas station in Burlington, Vermont into their first ice cream shop. Today, Ben & Jerry’s Homemade Holdings, Inc. has over 600 scoop shops in 16 countries. The company’s three-part mission statement emphasizes product quality, economic reward, and a commitment to the community. Ben & Jerry’s contributes a minimum of $1.1 million annually through corporate philanthropy that is primarily employee led.

I

xvii

Business Capsules appear at the ends of numerous sections. These capsules and their accompanying exercises deal with business situations that are related to the mathematical concepts covered in the chapter.

73. Research Project Use your school’s library, the Internet, or some other reference source to find information on a company that is noted for its philanthropy and community commitment. (One such business is described above.) Write a short paper about the company.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

xviii

Features

196

CHAPTER 2

Differentiation

Algebra Review Simplifying Algebraic Expressions To be successful in using derivatives, you must be good at simplifying algebraic expressions. Here are some helpful simplification techniques.

ALGEBRA REVIEWS

TECHNOLOGY

These appear throughout each chapter and offer algebraic support at point of use. Many of the reviews are then revisited in the Algebra Review at the end of the chapter, where additional details of examples with solutions and explanations are provided.

Symbolic algebra systems can simplify algebraic expressions. If you have access to such a system, try using it to simplify the expressions in this Algebra Review.

1. Combine like terms. This may involve expanding an expression by multiplying factors. 2. Divide out like factors in the numerator and denominator of an expression. 3. Factor an expression. 4. Rationalize a denominator. 5. Add, subtract, multiply, or divide fractions.

Example 1

Simplifying a Fractional Expression

共x ⫹ ⌬x兲2 ⫺ x 2 x 2 ⫹ 2x共⌬x兲 ⫹ 共⌬x兲2 ⫺ x2 ⫽ a. ⌬x ⌬x 2x共⌬x兲 ⫹ 共⌬x兲2 ⌬x

Combine like terms.



⌬x共2x ⫹ ⌬x兲 ⌬x

Factor.

⫽ 2x ⫹ ⌬x, b.

STUDY TIP When differentiating functions involving radicals, you should rewrite the function with rational exponents. For instance, you 3 x as should rewrite y ⫽ 冪 1兾3 y ⫽ x , and you should rewrite y⫽

1 3 x4 冪

⌬x ⫽ 0

Divide out like factors.

共x 2 ⫺ 1兲共⫺2 ⫺ 2x兲 ⫺ 共3 ⫺ 2x ⫺ x 2兲共2兲 共x 2 ⫺ 1兲2

Algebra Review For help in evaluating the expressions in Examples 3–6, see the review of simplifying fractional expressions on page 196.

Expand expression.





共⫺2x 2 ⫺ 2x 3 ⫹ 2 ⫹ 2x兲 ⫺ 共6 ⫺ 4x ⫺ 2x 2兲 共x 2 ⫺ 1兲2

Expand expression.



⫺2x 2 ⫺ 2x 3 ⫹ 2 ⫹ 2x ⫺ 6 ⫹ 4x ⫹ 2x 2 共x 2 ⫺ 1兲2

Remove parentheses.



⫺2x 3 ⫹ 6x ⫺ 4 共x 2 ⫺ 1兲2

Combine like terms.

2x ⫹ 1 c. 2 3x



⫽2

冣冤

3x共2兲 ⫺ 共2x ⫹ 1兲共3兲 共3x兲2



冢2x3x⫹ 1冣冤 6x ⫺共3x共6x兲 ⫹ 3兲冥 2

Multiply factors.



2共2x ⫹ 1兲共6x ⫺ 6x ⫺ 3兲 共3x兲3

Multiply fractions and remove parentheses.



2共2x ⫹ 1兲共⫺3兲 3共9兲x 3

Combine like terms and factor.



⫺2共2x ⫹ 1兲 9x 3

Divide out like factors.

STUDY TIPS Scattered throughout the text, study tips address special cases, expand on concepts, and help you to avoid common errors.

as y ⫽ x⫺4兾3. STUDY TIP In real-life problems, it is important to list the units of measure for a rate of change. The units for ⌬y兾⌬x are “y-units” per “x-units.” For example, if y is measured in miles and x is measured in hours, then ⌬y兾⌬x is measured in miles per hour.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

xix

Features

SECTION 2.3

Skills Review 2.3

These exercises at the beginning of each exercise set help students review skills covered in previous sections. The answers are provided at the back of the text to reinforce understanding of the skill sets learned.

In Exercises 1 and 2, evaluate the expression.

CHAPTER 2

Number of visitors (in hundreds of thousands)

V

where v is the wind speed (in meters per second). (a) Find

1200 900 600 300 t

Month (1 ↔ January)

(a) Estimate the rate of change of V over the interval 关9, 12兴 and explain your results. (b) Over what interval is the average rate of change approximately equal to the rate of change at t ⫽ 8? Explain your reasoning. 14. Medicine The graph shows the estimated number of milligrams of a pain medication M in the bloodstream t hours after a 1000-milligram dose of the drug has been given. Pain Medication in Bloodstream

Pain medication (in milligrams)

M

1000 800 600 400 200 4

5

6

7

t

Hours

(a) Estimate the one-hour interval over which the average rate of change is the greatest. (b) Over what interval is the average rate of change approximately equal to the rate of change at t ⫽ 4? Explain your reasoning. 15. Medicine The effectiveness E (on a scale from 0 to 1) of a pain-killing drug t hours after entering the bloodstream is given by E⫽

1 共9t ⫹ 3t 2 ⫺ t 3兲, 27

0 ≤ t ≤ 4.5.

Find the average rate of change of E on each indicated interval and compare this rate with the instantaneous rates of change at the endpoints of the interval. (a) 关0, 1兴

(b) 关1, 2兴

(c) 关2, 3兴

1 8. y ⫽ 9共6x 3 ⫺ 18x 2 ⫹ 63x ⫺ 15兲

x2 5000

10. y ⫽ 138 ⫹ 74x ⫺

(d) 关3, 4兴

(b) 1985–1990

(c) 1990–1995

(d) 1995–2000

(e) 1980–2004

(f) 1990–2004

t

0

1

2

3

4

5

6

A

63

72

81

90

102

115

120

t

7

t dH and interpret its meaning in this situation. dv A

8

9

10

11

12

134

142

152

161

165

14

15

16

17

18

166

169

184

197

212

228

19

20

21

22

23

24

(c) Imports: 1990–2000

(d) Exports: 1990–2000

(e) Imports: 1980–2005

(f) Exports: 1980–2005

Trade Deficit 1800

Value of goods (in billions of dollars)

(a) 1980–1985

A

x3 10,000

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1. Research and Development The table shows the amounts A (in billions of dollars per year) spent on R&D in the United States from 1980 through 2004, where t is the year, with t ⫽ 0 corresponding to 1980. Approximate the average rate of change of A during each period. (Source: U.S. National Science Foundation)

A 126 16. Chemistry: Wind Chill At 0⬚ Celsius, the heat loss H (in kilocalories per square meter per hour) from a person’s 13 t body can be modeled by

1500

3

6. y ⫽ ⫺16x 2 ⫹ 54x ⫹ 70

9. y ⫽ 12x ⫺

H ⫽ 33共10冪v ⫺ v ⫹ 10.45兲

Visitors to a National Park

2

4. y ⫽ ⫺3t 3 ⫹ 2t 2 ⫺ 8

5. s ⫽ ⫺16t 2 ⫹ 24t ⫹ 30

Exercises 2.3

13. Consumer Trends The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period, where t ⫽ 1 represents January.

1

⫺37 ⫺ 54 16 ⫺ 3

3. y ⫽ 4x 2 ⫺ 2x ⫹ 7 1 7. A ⫽ 10共⫺2r3 ⫹ 3r 2 ⫹ 5r兲

Differentiation

1 2 3 4 5 6 7 8 9 10 11 12

2.

I

1600 1400 1200 1000

E

800 600 400 200 5

10

15

20

25

30

t

Year (0 ↔ 1980) Figure for 2

In Exercises 3–12, use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval.

3. f 共t兲 ⫽ 3t ⫹ 5; 关1, 2兴 4. h共x兲 ⫽ 2 ⫺ x; 关0, 2兴 312 5. h共x兲 ⫽ x 2 ⫺ 4x ⫹ 2; 关⫺2, 2兴 (b) Find the rates of change of H when v ⫽ 2 and when 2 v ⫽ 5. 2. Trade Deficit The graph shows the values I (in billions 6. f 共x兲 ⫽ x ⫺ 6x ⫺ 1; 关⫺1, 3兴 of of dollars per year) of goods imported to the United States 17. Velocity The height s (in feet) at time t (in seconds) a 7. f (x) ⫽ 3x4兾3; 关1, 8兴 8. f 共x兲 ⫽ x3兾2; 关1, 4] and the values CEH(in per year) of goods silver dollar dropped from the top of the Washington 152 A Pbillions T E R 2of dollars Differentiation 1 1 exported from the United States from 1980 through Monument is given by ; 关1, 4兴 9. f 共x兲 ⫽ ; 关1, 4兴 10. f 共x兲 ⫽ x 2005. each indicated rate of change. 46. Gasoline Sales The number N 冪 ofx gallons of regular C of producing x units is 40. Approximate Marginal Cost The cost average s ⫽ ⫺16t 2 ⫹ 555. (Source: U.S. International Administration) unleaded ⫹ k, where modeled by C ⫽ v共x兲 Trade v represents the variable 11. g共x兲 ⫽ x 4 ⫺ x 2gasoline ⫹ 2; 关1,sold 3兴 by a gasoline station at a price of p (a) Find the average velocity on the interval 关2, 3兴. (a) Imports: dollars per gallon is given by N ⫽ f 共p兲. k represents the(b) cost and1980–1990 fixed cost. Show that the marginal Exports: 1980–1990 12. g共x兲 ⫽ x3 ⫺ 1; 关⫺1, 1兴 (b) Find the instantaneous velocities when t ⫽ 2 and whencost is independent of the fixed cost. (a) Describe the meaning of f⬘共2.959) t ⫽ 3. 41. Marginal Profit When the admission price for a (b) Is f⬘共2.959) usually positive or negative? Explain. baseball game was $6 per ticket, 36,000 tickets were sold. (c) How long will it take the dollar to hit the ground? 47. Dow Jones Industrial Average The table shows the When the price was raised to $7, only 33,000 tickets were (d) Find the velocity of the dollar when it hits the ground. year-end closing prices p of the Dow Jones Industrial sold. Assume that the demand function is linear and that the Average (DJIA) from 1992 through 2006, where t is the 18. Physics: Velocity A racecar travels northward on avariable and fixed costs for the ballpark owners are $0.20 year, and t ⫽ 2 corresponds to 1992. (Source: Dow straight, level track at a constant speed, traveling 0.750and $85,000, respectively. Jones Industrial Average) kilometer in 20.0 seconds. The return trip over the same (a) Find the profit P as a function of x, the number of track is made in 25.0 seconds. tickets sold. 2 3 4 5 6 t (a) What is the average velocity of the car in meters per (b) Use a graphing utility to graph P, and comment about second for the first leg of the run? p 3301.11 3754.09 3834.44 5117.12 6448.26 the slopes of P when x ⫽ 18,000 and when x ⫽ 36,000. (b) What is the average velocity for the total trip? (c) Find the marginal profits when 18,000 tickets are sold 8 9 10 11 t 7 (Source: Shipman/Wilson/Todd, An Introduction to Physiand when 36,000 tickets are sold. cal Science, Eleventh Edition) 42. Marginal Profit In Exercise 41, suppose ticket sales p 7908.24 9181.43 11,497.12 10,786.85 10,021.50 decreased to 30,000 when the price increased to $7. How Marginal Cost In Exercises 19–22, find the marginal would this change the answers? 13 14 15 16 t 12 cost for producing x units. (The cost is measured in 43. Profit The demand function for a product is given by dollars.) p 8341.63 10,453.92 10,783.01 10,717.50 12,463.15 p ⫽ 50兾冪x for 1 ≤ x ≤ 8000, and the cost function is 19. C ⫽ 4500 ⫹ 1.47x 20. C ⫽ 205,000 ⫹ 9800x given by C ⫽ 0.5x ⫹ 500 for 0 ≤ x ≤ 8000. 21. C ⫽ 55,000 ⫹ 470x ⫺ 0.25x 2, 0 ≤ x ≤ 940 (a) Determine the average rate of change in the value of the Find the marginal profits for (a) x ⫽ 900, (b) x ⫽ 1600, DJIA from 1992 to 2006. 22. C ⫽ 100共9 ⫹ 3冪x 兲 (c) x ⫽ 2500, and (d) x ⫽ 3600. (b) Estimate the instantaneous rate of change in 1998 by Marginal Revenue In Exercises 23–26, find theIf you were in charge of setting the price for this product, finding the average rate of change from 1996 to 2000. marginal revenue for producing x units. (The revenuewhat price would you set? Explain your reasoning. (c) Estimate the instantaneous rate of change in 1998 by is measured in dollars.) 44. Inventory Management The annual inventory cost finding the average rate of change from 1997 to 1999. for a manufacturer is given by 23. R ⫽ 50x ⫺ 0.5x 2 24. R ⫽ 30x ⫺ x 2 (d) Compare your answers for parts (b) and (c). Which C ⫽ 1,008,000兾Q ⫹ 6.3Q 25. R ⫽ ⫺6x 3 ⫹ 8x 2 ⫹ 200x 26. R ⫽ 50共20x ⫺ x3兾2兲 interval do you think produced the best estimate for the where Q is the order size when the inventory is replenished. instantaneous rate of change in 1998? Marginal Profit In Exercises 27–30, find the marginalFind the change in annual cost when Q is increased from profit for producing x units. (The profit is measured in350 to 351, and compare this with the instantaneous rate of 48. Biology Many populations in nature exhibit logistic dollars.) change when Q ⫽ 350. growth, which consists of four phases, as shown in the figure. Describe the rate of growth of the population in each 27. P ⫽ ⫺2x 2 ⫹ 72x ⫺ 145 45. MAKE A DECISION: FUEL COST A car is driven 15,000 phase, and give possible reasons as to why the rates might miles a year and gets x miles per gallon. Assume that the 28. P ⫽ ⫺0.25x 2 ⫹ 2000x ⫺ 1,250,000 be changing from phase to phase. (Source: Adapted from average fuel cost is $2.95 per gallon. Find the annual cost 2 29. P ⫽ ⫺0.00025x ⫹ 12.2x ⫺ 25,000 Levine/Miller, Biology: Discovering Life, Second Edition) of fuel C as a function of x and use this function to 30. P ⫽ ⫺0.5x 3 ⫹ 30x 2 ⫺ 164.25x ⫺ 1000 complete the table. 245

267

x

277

10

15

276

292

20

25

30

35

Acceleration Deceleration phase phase

40

C dC兾dx Who would benefit more from a 1 mile per gallon increase in fuel efficiency—the driver who gets 15 miles per gallon or the driver who gets 35 miles per gallon? Explain.

Lag phase

Population

150

⫺63 ⫺ 共⫺105兲 21 ⫺ 7

In Exercises 3–10, find the derivative of the function.

EXERCISE SETS These exercises offer opportunities for practice and review. They progress in difficulty from skill-development problems to more challenging problems, to build confidence and understanding.

149

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 2.1 and 2.2.

SKILLS REVIEW

1.

Rates of Change: Velocity and Marginals

Equilibrium

Time

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

xx

Features

164

CHAPTER 2

Differentiation

Mid-Chapter Quiz

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Take this quiz as you would take a quiz in class. When you are done, check your work against the answers given in the back of the book. In Exercises 1–3, use the limit definition to find the derivative of the function. Then find the slope of the tangent line to the graph of f at the given point.

MID-CHAPTER QUIZ

1. f 共x兲 ⫽ ⫺x ⫹ 2; 共2, 0兲

Appearing in the middle of each chapter, this one page test allows you to practice skills and concepts learned in the chapter. This opportunity for self-assessment will uncover any potential weak areas that might require further review of the material.

2. f 共x兲 ⫽ 冪x ⫹ 3; 共1, 2)

4 3. f 共x兲 ⫽ ; 共1, 4) x

In Exercises 4 –12, find the derivative of the function. 4. f (x) ⫽ 12

5. f 共x) ⫽ 19x ⫹ 9

7. f (x) ⫽ 12x1兾4

8. f (x) ⫽ 4x⫺2

10. f 共x兲 ⫽

2x ⫹ 3 3x ⫹ 2

11. f (x兲 ⫽ 共x2 ⫹ 1兲共⫺2x ⫹ 4)

6. f 共x兲 ⫽ 5 ⫺ 3x2 9. f (x) ⫽ 2冪x 12. f 共x兲 ⫽

4⫺x x⫹5

In Exercises 13–16, use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. 13. f 共x兲 ⫽ x2 ⫺ 3x ⫹ 1; 关0, 3兴 14. f 共x兲 ⫽ 2x3 ⫹ x2 ⫺ x ⫹ 4; 关⫺1, 1兴 15. f 共x兲 ⫽

1 ; [2, 5兴 2x

3 x; 关8, 27兴 16. f 共x兲 ⫽ 冪

17. The profit (in dollars) from selling x units of a product is given by P ⫽ ⫺0.0125x2 ⫹ 16x ⫺ 600 (a) Find the additional profit when the sales increase from 175 to 176 units. (b) Find the marginal profit when x ⫽ 175. (c) Compare the results of parts (a) and (b). In Exercises 18 and 19, find an equation of the tangent line to the graph of f at the given point. Then use a graphing utility to graph the function and the equation of the tangent line in the same viewing window. 18. f 共x) ⫽ 5x2 ⫹ 6x ⫺ 1; 共⫺1, ⫺2兲 19. f (x兲 ⫽ 共x ⫺ 1兲共x ⫹ 1); 共0, ⫺1兲 20. From 2000 through 2005, the sales per share S (in dollars) for CVS Corporation can be modeled by S ⫽ 0.18390t 3 ⫺ 0.8242t2 ⫹ 3.492t ⫹ 25.60, 0 ≤ t ≤ 5

204

CHAPTER 2

where t represents the year, with t ⫽ 0 corresponding to 2000. Corporation)

Differentiation

(Source: CVS

(a) Find the rate of change of the sales per share with respect to the year.

Chapter Test

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

(b) At what rate were the sales per share changing in 2001? in 2004? in 2005?

Take this test as you would take a test in class. When you are done, check your work against the answers given in the back of the book. In Exercises 1 and 2, use the limit definition to find the derivative of the function. Then find the slope of the tangent line to the graph of f at the given point. 1. f 共x兲 ⫽ x2 ⫹ 1; 共2, 5兲

2. f 共x兲 ⫽ 冪x ⫺ 2; 共4, 0兲

In Exercises 3 –11, find the derivative of the function. Simplify your result. 3. f 共t兲 ⫽ t3 ⫹ 2t

4. f 共x兲 ⫽ 4x2 ⫺ 8x ⫹ 1

6. f 共x兲 ⫽ 共x ⫹ 3兲共x ⫺ 3兲

7. f 共x兲 ⫽ ⫺3x⫺3

9. f 共x兲 ⫽ 共3x2 ⫹ 4兲2

10. f 共x兲 ⫽ 冪1 ⫺ 2x

5. f 共x兲 ⫽ x3兾2 8. f 共x兲 ⫽ 冪x 共5 ⫹ x兲 11. f 共x兲 ⫽

共5x ⫺ 1兲3 x

1 at the point 共1, 0兲. x Then use a graphing utility to graph the function and the tangent line in the same viewing window.

12. Find an equation of the tangent line to the graph of f 共x兲 ⫽ x ⫺

13. The annual sales S (in millions of dollars per year) of Bausch & Lomb for the years 1999 through 2005 can be modeled by S ⫽ ⫺2.9667t 3 ⫹ 135.008t 2 ⫺ 1824.42t ⫹ 9426.3, 9 ≤ t ≤ 15 where t represents the year, with t ⫽ 9 corresponding to 1999. Lomb, Inc.)

(Source: Bausch &

(a) Find the average rate of change for the interval from 2001 through 2005. (b) Find the instantaneous rates of change of the model for 2001 and 2005.

CHAPTER TEST Appearing at the end of the chapter, this test is designed to simulate an in-class exam. Taking these tests will help you to determine what concepts require further study and review.

(c) Interpret the results of parts (a) and (b) in the context of the problem. 14. The monthly demand and cost functions for a product are given by p ⫽ 1700 ⫺ 0.016x

and

C ⫽ 715,000 ⫹ 240x.

Write the profit function for this product. In Exercises 15–17, find the third derivative of the function. Simplify your result. 15. f 共x兲 ⫽ 2x2 ⫹ 3x ⫹ 1

16. f 共x兲 ⫽ 冪3 ⫺ x

17. f 共x兲 ⫽

2x ⫹ 1 2x ⫺ 1

In Exercises 18–20, use implicit differentiation to find dy/dx. 18. x ⫹ xy ⫽ 6

19. y2 ⫹ 2x ⫺ 2y ⫹ 1 ⫽ 0

20. x2 ⫺ 2y2 ⫽ 4

21. The radius r of a right circular cylinder is increasing at a rate of 0.25 centimeter per minute. The height h of the cylinder is related to the radius by h ⫽ 20r. Find the rate of change of the volume when (a) r ⫽ 0.5 centimeter and (b) r ⫽ 1 centimeter.

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Features

198

CHAPTER 2

Differentiation

Chapter Summary and Study Strategies After studying this chapter, you should have acquired the following skills. The exercise numbers are keyed to the Review Exercises that begin on page 200. Answers to odd-numbered Review Exercises are given in the back of the text.*

C H A P T E R S U M M A RY A N D S T U D Y S T R AT E G I E S The Chapter Summary reviews the skills covered in the chapter and correlates each skill to the Review Exercises that test the skill. Following each Chapter Summary is a short list of Study Strategies for addressing topics or situations in the chapter.

Section 2.1

Review Exercises 1– 4



Approximate the slope of the tangent line to a graph at a point.



Interpret the slope of a graph in a real-life setting.

5– 8



Use the limit definition to find the derivative of a function and the slope of a graph at a point.

9–16



Use the derivative to find the slope of a graph at a point.

17–24



Use the graph of a function to recognize points at which the function is not differentiable.

25–28

f⬘共x兲 ⫽ lim

⌬x→0

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

Section 2.2 ■

Use the Constant Multiple Rule for differentiation.

29, 30

d 关c f 共x兲兴 ⫽ c f⬘共x兲 dx ■

Use the Sum and Difference Rules for differentiation.

31–38

d 关 f 共x兲 ± g共x兲兴 ⫽ f⬘共x兲 ± g⬘共x兲 dx

Section 2.3 ■

Find the average rate of change of a function over an interval and the instantaneous rate of change at a point.

39, 40

f 共b兲 ⫺ f 共a兲 Average rate of change ⫽ b⫺a Instantaneous rate of change ⫽ lim

⌬x→0

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x



Find the average and instantaneous rates of change of a quantity in a real-life problem.

41–44



Find the velocity of an object that is moving in a straight line.

45, 46



Create mathematical models for the revenue, cost, and profit for a product.

47, 48

P ⫽ R ⫺ C, R ⫽ xp ■

Applications Business and Economics Account balances, 302 Advertising costs, 195 Annual operating costs, 7 Annual salary, 68 Annuity, 18, 390, 393, 415 Average cost, 233, 252, 261, 265, 388 Average cost and profit, 289 Average production, 560 Average profit, 265, 558 Average revenue, 560 Average salary for public school nurses, 380 Average weekly profit, 560 Bolts produced by a foundry, 381 Break-even analysis, 54, 68, 110 Break-even point, 49, 55 Budget deficit, 401 Budget variance, 12 Capital accumulation, 393 Capital campaign, 428 Capitalized cost, 469, 476 Cash flow, 373 Cash flow per share Energizer Holdings, 62 Harley-Davidson, 339 Ruby Tuesday, 62 Certificate of deposit, 307 Charitable foundation, 469 Choosing a job, 67 Cobb-Douglas production function, 187, 500, 503, 514, 528, 560 College tuition fund, 428 Compact disc shipments, 287 Complementary and substitute products, 514 Compound interest, 18, 93, 101, 104, 173, 306, 315, 316, 324, 338, 342, 349, 393, 415 Construction, 41, 534 Consumer and producer surplus, 398, 401, 402, 416, 417, 448 Cost, 58, 80, 81, 99, 137, 163, 214, 224, 265, 274, 361, 363, 364, 373, 393, 413, 414, 524, 533 Cost, revenue, and profit, 81, 194, 202, 402 Pixar, 109 Credit card rate, 173 Daily morning newspapers, number of, 541 Demand, 80, 110, 145, 146, 151, 152, 162, 163, 185, 187, 254, 282, 290, 306, 324, 333, 348, 363, 380, 427, 543 Demand function, 373, 509

Depreciation, 64, 67, 110, 173, 298, 315, 351, 393 Diminishing returns, 231, 244 Doubling time, 322, 324, 352 Dow Jones Industrial Average, 41, 152, 234 Earnings per share Home Depot, 477 Starbucks, 504 Earnings per share, sales, and shareholder’s equity, PepsiCo, 544 Economics, 151 equation of exchange, 566 gross domestic product, 282 marginal benefits and costs, 364 present value, 474 revenue, 290 Economy, contour map, 499 Effective rate of interest, 303, 306, 349 Effective yield, 342 Elasticity of demand, 253 Elasticity and revenue, 250 Endowment, 469 Equilibrium point, 50, 113 Equimarginal Rule, 533 Farms, number of, 113 Federal education spending, 55 Finance, 24, 325 present value, 474 Fuel cost, 152, 399 Future value, 306, 428 Hourly wage, 350, 539 Income median, 543 personal, 67 Income distribution, 402 Increasing production, 193 Inflation rate, 298, 316, 351 Installment loan, 32 Interval of inelasticity, 291 Inventory, 32 cost, 233, 289 management, 104, 152 replenishment, 163 Investment, 504, 515 Rule of 70, 342 strategy, 534 Job offer, 401 Least-Cost Rule, 533 Linear depreciation, 64, 66, 67, 110 Lorenz curve, 402 Managing a store, 163 Manufacturing, 12 Marginal analysis, 277, 278, 282, 393, 457 Marginal cost, 150, 151, 152, 202, 381, 514, 567

Find the marginal revenue, marginal cost, and marginal profit for a product.

49–58

* Use a wide range of valuable study aids to help you master the material in this chapter. The Student Solutions Guide includes step-by-step solutions to all odd-numbered exercises to help you review and prepare. The student website at college.hmco.com/info/larsonapplied offers algebra help and a Marginal productivity, 514 Graphing Technology Guide. The Graphing Technology Guide contains step-by-step commands and instructions for a wide variety of graphing calculators, including the most recent models. Marginal profit, 144, 148, 150, 151,

152, 164, 202, 203 Marginal revenue, 147, 150, 151, 202, 514, 567 Market equilibrium, 81 Marketing, 437 Maximum production level, 528, 529, 567, 569 Maximum profit, 222, 248, 252, 253, 520, 530 Maximum revenue, 245, 247, 253, 312 Minimum average cost, 246, 333, 334 Minimum cost, 241, 242, 243, 253, 288, 525, 567 Monthly payments, 501, 504 Mortgage debt, 393 National debt, 112 Negotiating a price, 162 Number of Kohl’s stores, 449 Office space, 534 Owning a business, 80 a franchise, 104 Point of diminishing returns, 231, 233, 288 Present value, 304, 306, 349, 424, 425, 428, 449, 457, 469, 474, 476 of a perpetual annuity, 467 Producer and consumer surplus, 398, 401, 402, 416, 417, 448 Production, 12, 187, 413, 500, 503, 533 Production level, 6, 24 Productivity, 233 Profit, 7, 24, 67, 81, 93, 104, 151, 152, 164, 192, 195, 202, 203, 204, 214, 224, 243, 274, 281, 288, 289, 343, 364, 387, 415, 503, 523, 567 Affiliated Computer Services, 351 Bank of America, 351 CVS Corporation, 42 The Hershey Company, 448 Walt Disney Company, 42 Profit analysis, 67, 212, 214 Property value, 298, 348 Purchasing power of the dollar, 448 Quality control, 11, 12, 162, 469 Real estate, 80, 568 Reimbursed expenses, 68 Retail values of motor homes, 180 Revenue, 81, 150, 151, 254, 281, 288, 343, 380, 401, 413, 428, 448, 523, 567 California Pizza Kitchen, 348 CVS Corporation, 42 EarthLink, 544

A P P L I C AT I O N I N D E X This list, found on the front endsheets, is an index of all the applications presented in the text Examples and Exercises.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

xxi

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0

AP/Wide World Photos

A Precalculus Review

The annual operating costs of each van owned by a utility company can be determined by solving an inequality. (See Section 0.1, Exercise 36.)

0.1 0.2

Applications Topics in precalculus have many real-life applications. The applications listed below represent a sample of the applications in this chapter. ■ ■ ■ ■

Sales, Exercise 35, page 7 Quality Control, Exercise 51, page 12 Production Level, Exercise 75, page 24 Make a Decision: Inventory, Exercise 48, page 32

0.3 0.4 0.5

The Real Number Line and Order Absolute Value and Distance on the Real Number Line Exponents and Radicals Factoring Polynomials Fractions and Rationalization

1 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

2

CHAPTER 0

A Precalculus Review

Section 0.1 ■ Represent, classify, and order real numbers.

The Real Number Line and Order

■ Use inequalities to represent sets of real numbers. ■ Solve inequalities. ■ Use inequalities to model and solve real-life problems.

The Real Number Line Real numbers can be represented with a coordinate system called the real number line (or x-axis), as shown in Figure 0.1. The positive direction (to the right) is denoted by an arrowhead and indicates the direction of increasing values x of x. The real number corresponding to a particular point on the real number line − 4 − 3 −2 − 1 0 1 2 3 4 F I G U R E 0 . 1 The Real Number Line is called the coordinate of the point. As shown in Figure 0.1, it is customary to label those points whose coordinates are integers. The point on the real number line corresponding to zero is called the origin. Numbers to the right of the origin are positive, and numbers to the left of the origin are negative. The term nonnegative describes a number that is either positive or zero. 5 The importance of the real number line is that it provides you with a − 2.6 4 conceptually perfect picture of the real numbers. That is, each point on the real number line corresponds to one and only one real number, and each real number x corresponds to one and only one point on the real number line. This type of rela−3 −2 −1 0 1 2 3 tionship is called a one-to-one correspondence and is illustrated in Figure 0.2. Every point on the real number line Each of the four points in Figure 0.2 corresponds to a real number that can corresponds to one and only one real number. be expressed as the ratio of two integers. Negative direction (x decreases)

Positive direction (x increases)

7

−3

−3

1.85

−2

−1

0

1

⫺2.6 ⫽ ⫺ 13 5

2

3

x

Every real number corresponds to one and only one point on the real number line.

2

−1

0

FIGURE 0.3

1

e

2

π

3

x

⫺ 73

1.85 ⫽

37 20

Such numbers are called rational. Rational numbers have either terminating or infinitely repeating decimal representations. Terminating Decimals 2 ⫽ 0.4 5 7 ⫽ 0.875 8

FIGURE 0.2

5 4

Infinitely Repeating Decimals 1 ⫽ 0.333 . . . ⫽ 0.3* 3 12 ⫽ 1.714285714285 . . . ⫽ 1.714285 7

Real numbers that are not rational are called irrational, and they cannot be represented as the ratio of two integers (or as terminating or infinitely repeating decimals). So, a decimal approximation is used to represent an irrational number. Some irrational numbers occur so frequently in applications that mathematicians have invented special symbols to represent them. For example, the symbols 冪2, ␲, and e represent irrational numbers whose decimal approximations are as shown. (See Figure 0.3.) 冪2 ⬇ 1.4142135623

␲ ⬇ 3.1415926535

e ⬇ 2.7182818284

*The bar indicates which digit or digits repeat infinitely.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 0.1

3

The Real Number Line and Order

Order and Intervals on the Real Number Line One important property of the real numbers is that they are ordered: 0 is less 22 than 1, ⫺3 is less than ⫺2.5, ␲ is less than 7 , and so on. You can visualize this property on the real number line by observing that a is less than b if and only if a lies to the left of b on the real number line. Symbolically, “a is less than b” is 3 denoted by the inequality a < b. For example, the inequality 4 < 1 follows from 3 the fact that 4 lies to the left of 1 on the real number line, as shown in Figure 0.4. 3 4

lies to the left of 1, so 3 4

−1

0

3 4

< 1.

1 1

2

x

FIGURE 0.4

When three real numbers a, x, and b are ordered such that a < x and x < b, we say that x is between a and b and write a < x < b.

x is between a and b.

The set of all real numbers between a and b is called the open interval between a and b and is denoted by 共a, b兲. An interval of the form 共a, b兲 does not contain the “endpoints” a and b. Intervals that include their endpoints are called closed and are denoted by 关a, b兴. Intervals of the form 关a, b兲 and 共a, b兴 are neither open nor closed. Figure 0.5 shows the nine types of intervals on the real number line.

Intervals that are neither open nor closed

Open interval

a

b

a

a b

(b, ∞) a

b

b

x≤a

[b, ∞) a

b

b

x≥b

a≤xb

(−∞, a] a

[a, b]

FIGURE 0.5

a

x bc, c < 0

5. Adding a constant: a < b 6. Subtracting a constant: a < b

a⫹c < b⫹c a⫺c < b⫺c

Note that you reverse the inequality when you multiply by a negative number. For example, if x < 3, then ⫺4x > ⫺12. This principle also applies to division by a negative number. So, if ⫺2x > 4, then x < ⫺2.

Example 1

Solving an Inequality an Inequality

Find the solution set of the inequality 3x ⫺ 4 < 5. SOLUTION

For x = 0, 3(0) − 4 = − 4. For x = 2, 3(2) − 4 = 2. For x = 4, 3(4) − 4 = 8. x

−1

0

1

2

Solution set for 3x − 4 < 5

FIGURE 0.6

3

4

5

6

7

8

3x ⫺ 4 < 5 3x ⫺ 4 ⫹ 4 < 5 ⫹ 4 3x < 9 1 1 共3x兲 < 共9兲 3 3 x < 3

Write original inequality. Add 4 to each side. Simplify. 1

Multiply each side by 3 . Simplify.

So, the solution set is the interval 共⫺ ⬁, 3兲, as shown in Figure 0.6.

✓CHECKPOINT 1 Find the solution set of the inequality 2x ⫺ 3 < 7.



In Example 1, all five inequalities listed as steps in the solution have the same solution set, and they are called equivalent inequalities.

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SECTION 0.1

The Real Number Line and Order

5

The inequality in Example 1 involves a first-degree polynomial. To solve inequalities involving polynomials of higher degree, you can use the fact that a polynomial can change signs only at its real zeros (the real numbers that make the polynomial zero). Between two consecutive real zeros, a polynomial must be entirely positive or entirely negative. This means that when the real zeros of a polynomial are put in order, they divide the real number line into test intervals in which the polynomial has no sign changes. That is, if a polynomial has the factored form

共x ⫺ r1兲共x ⫺ r2兲, . . . , 共x ⫺ rn 兲,

r1 < r2 < r3 < . . . < rn

then the test intervals are

共⫺ ⬁, r1兲, 共r1, r2兲, . . . , 共rn⫺1, rn 兲, and 共rn, ⬁兲. For example, the polynomial x2 ⫺ x ⫺ 6 ⫽ 共x ⫺ 3兲共x ⫹ 2兲 can change signs only at x ⫽ ⫺2 and x ⫽ 3. To determine the sign of the polynomial in the intervals 共⫺ ⬁, ⫺2兲, 共⫺2, 3兲, and 共3, ⬁兲, you need to test only one value from each interval. Sign of 冇x ⴚ 3冈冇x 1 2冈

Example 2

x

Sign

< 0?

⫺3

共 ⫺ 兲共 ⫺ 兲

No

⫺2

共 ⫺ 兲共0兲

No

⫺1

共 ⫺ 兲共 ⫹ 兲

Yes

0

共 ⫺ 兲共 ⫹ 兲

Yes

1

共 ⫺ 兲共 ⫹ 兲

Yes

2

共 ⫺ 兲共 ⫹ 兲

Yes

3

共0兲共 ⫹ 兲

No

4

共 ⫹ 兲共 ⫹ 兲

No

Find the solution set of the inequality x2 < x ⫹ 6. SOLUTION

x2 < x ⫹ 6 x2 ⫺ x ⫺ 6 < 0 共x ⫺ 3兲共x ⫹ 2兲 < 0

No (−)(−) > 0

FIGURE 0.7

Write original inequality. Polynomial form Factor.

So, the polynomial x2 ⫺ x ⫺ 6 has x ⫽ ⫺2 and x ⫽ 3 as its zeros. You can solve the inequality by testing the sign of the polynomial in each of the following intervals. x < ⫺2, ⫺2 < x < 3, x > 3 To test an interval, choose a representative number in the interval and compute the sign of each factor. For example, for any x < ⫺2, both of the factors 共x ⫺ 3兲 and 共x ⫹ 2兲 are negative. Consequently, the product (of two negative numbers) is positive, and the inequality is not satisfied in the interval x

−2

Solving a Polynomial Inequality Inequality

3

Yes No (−)(+) < 0 (+)(+) > 0

Is 共x ⫺ 3兲共x ⫹ 2兲 < 0?

x < ⫺2. A convenient testing format is shown in Figure 0.7. Because the inequality is satisfied only by the center test interval, you can conclude that the solution set is given by the interval ⫺2 < x < 3.

Solution set

✓CHECKPOINT 2 Find the solution set of the inequality x2 > 3x ⫹ 10.



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6

CHAPTER 0

A Precalculus Review

Application Inequalities are frequently used to describe conditions that occur in business and science. For instance, the inequality 144 ≤ W ≤ 180 describes the recommended weight W for a man whose height is 5 feet 10 inches. Example 3 shows how an inequality can be used to describe the production levels in a manufacturing plant.

Example 3

Production Levels

In addition to fixed overhead costs of $500 per day, the cost of producing x units of an item is $2.50 per unit. During the month of August, the total cost of production varied from a high of $1325 to a low of $1200 per day. Find the high and low production levels during the month. Because it costs $2.50 to produce one unit, it costs 2.5x to produce x units. Furthermore, because the fixed cost per day is $500, the total daily cost of producing x units is C ⫽ 2.5x ⫹ 500. Now, because the cost ranged from $1200 to $1325, you can write the following. SOLUTION

1200 1200 ⫺ 500 700 700 2.5 280

≤ ≤ ≤ ≤ ≤

2.5x ⫹ 500 2.5x ⫹ 500 ⫺ 500 2.5x 2.5x 2.5 x

≤ 1325

Write original inequality.

≤ 1325 ⫺ 500 Subtract 500 from each part. ≤ 825

825 2.5 ≤ 330 ≤

Simplify. Divide each part by 2.5. Simplify.

So, the daily production levels during the month of August varied from a low of 280 units to a high of 330 units, as shown in Figure 0.8. Each day’s production during the month fell in this interval. Low daily production 280 0

100

200

300

High daily production 330 400

x

500

FIGURE 0.8

✓CHECKPOINT 3 Use the information in Example 3 to find the high and low production levels if, during October, the total cost of production varied from a high of $1500 to a low of $1000 per day. ■ The symbol

indicates an example that uses or is derived from real-life data.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 0.1

Exercises 0.1

7

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–10, determine whether the real number is rational or irrational. 2. ⫺3678

*1. 0.25

The Real Number Line and Order

30. The estimated daily oil production p at a refinery is greater than 2 million barrels but less than 2.4 million barrels. 31. According to a survey, the percent p of Americans that now conduct most of their banking transactions online is no more than 40%.

3␲ 3. 2

4. 3冪2 ⫺ 1

5. 4.3451

6.

22 7

32. The net income I of a company is expected to be no less than $239 million.

3 64 7. 冪

8. 0.8177

33. Physiology The maximum heart rate of a person in normal health is related to the person’s age by the equation

3 60 9. 冪

10. 2e

In Exercises 11–14, determine whether each given value of x satisfies the inequality. 11. 5x ⫺ 12 > 0 (a) x ⫽ 3 12. x ⫹ 1
x⫹1 > 4 4 x x 25. ⫹ > 5 2 3 23.

27. 2x 2 ⫺ x < 6

24. ⫺1 < ⫺ 26.

x < 1 3

x x ⫺ > 5 2 3

28. 2x2 ⫹ 1 < 9x ⫺ 3

r ⫽ 220 ⫺ A where r is the maximum heart rate in beats per minute and A is the person’s age in years. Some physiologists recommend that during physical activity a person should strive to increase his or her heart rate to at least 60% of the maximum heart rate for sedentary people and at most 90% of the maximum heart rate for highly fit people. Express as an interval the range of the target heart rate for a 20-year-old. 34. Profit The revenue for selling x units of a product is R ⫽ 115.95x, and the cost of producing x units is C ⫽ 95x ⫹ 750. To obtain a profit, the revenue must be greater than the cost. For what values of x will this product return a profit? 35. Sales A doughnut shop at a shopping mall sells a dozen doughnuts for $4.50. Beyond the fixed cost (for rent, utilities, and insurance) of $220 per day, it costs $2.75 for enough materials (flour, sugar, etc.) and labor to produce each dozen doughnuts. If the daily profit varies between $60 and $270, between what levels (in dozens) do the daily sales vary? 36. Annual Operating Costs A utility company has a fleet of vans. The annual operating cost C (in dollars) of each van is estimated to be C ⫽ 0.35m ⫹ 2500, where m is the number of miles driven. The company wants the annual operating cost of each van to be less than $13,000. To do this, m must be less than what value? In Exercises 37 and 38, determine whether each statement is true or false, given a < b. 37. (a) ⫺2a < ⫺2b

38. (a) a ⫺ 4 < b ⫺ 4

In Exercises 29–32, use inequality notation to describe the subset of real numbers.

(b) a ⫹ 2 < b ⫹ 2

(b) 4 ⫺ a < 4 ⫺ b

(c) 6a < 6b

(c) ⫺3b < ⫺3a

29. A company expects its earnings per share E for the next quarter to be no less than $4.10 and no more than $4.25.

1 1 (d) < a b

(d)

a b < 4 4

*The answers to the odd-numbered and selected even-numbered exercises are given in the back of the text. Worked-out solutions to the odd-numbered exercises are given in the Student Solutions Guide.

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8

CHAPTER 0

A Precalculus Review

Section 0.2

Absolute Value and Distance on the Real Number Line

■ Find the absolute values of real numbers and understand the properties

of absolute value. ■ Find the distance between two numbers on the real number line. ■ Define intervals on the real number line. ■ Find the midpoint of an interval and use intervals to model and solve

real-life problems.

Absolute Value of a Real Number TECHNOLOGY Absolute value expressions can be evaluated on a graphing utility. When an expression such as 3 ⫺ 8 is evaluated, parentheses should surround the expression, as in abs共3 ⫺ 8兲.





Definition of Absolute Value

The absolute value of a real number a is

ⱍaⱍ ⫽ 冦⫺a, a,

if a ≥ 0 if a < 0.

At first glance, it may appear from this definition that the absolute value of a real number can be negative, but this is not possible. For example, let a ⫽ ⫺3. Then, because ⫺3 < 0, you have

ⱍaⱍ ⫽ ⱍ⫺3ⱍ ⫽ ⫺ 共⫺3兲 ⫽ 3. The following properties are useful for working with absolute values. Properties of Absolute Value

ⱍ ⱍ ⱍ ⱍⱍ ⱍ a a ⫽ ⱍ ⱍ, b ⫽ 0 b ⱍbⱍ ⱍanⱍ ⫽ ⱍaⱍn 冪a2 ⫽ ⱍaⱍ

1. Multiplication: ab ⫽ a b 2. Division: 3. Power: 4. Square root:

ⱍⱍ

Be sure you understand the fourth property in this list. A common error in algebra is to imagine that by squaring a number and then taking the square root, you come back to the original number. But this is true only if the original number is nonnegative. For instance, if a ⫽ 2, then 冪22 ⫽ 冪4 ⫽ 2

but if a ⫽ ⫺2, then 冪共⫺2兲2 ⫽ 冪4 ⫽ 2.

The reason for this is that (by definition) the square root symbol 冪 denotes only the nonnegative root.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 0.2

9

Absolute Value and Distance on the Real Number Line

Distance on the Real Number Line Directed distance from a to b:

Consider two distinct points on the real number line, as shown in Figure 0.9. b

a

x

2. The directed distance from b to a is a ⫺ b.

b−a



b

a

x

a−b Distance between a and b: b

⏐a − b⏐ or ⏐b − a⏐

FIGURE 0.9

ⱍ ⱍ



3. The distance between a and b is a ⫺ b or b ⫺ a .

Directed distance from b to a:

a

1. The directed distance from a to b is b ⫺ a.

x

In Figure 0.9, note that because b is to the right of a, the directed distance from a to b (moving to the right) is positive. Moreover, because a is to the left of b, the directed distance from b to a (moving to the left) is negative. The distance between two points on the real number line can never be negative. Distance Between Two Points on the Real Number Line

The distance d between points x1 and x2 on the real number line is given by





d ⫽ x2 ⫺ x1 ⫽ 冪共x2 ⫺ x1兲2 . Note that the order of subtraction with x1 and x2 does not matter because

ⱍx2 ⫺ x1ⱍ ⫽ ⱍx1 ⫺ x2ⱍ Example 1

共x2 ⫺ x1兲2 ⫽ 共x1 ⫺ x2 兲2.

and

Finding Distance on the Real Number Line

Determine the distance between ⫺3 and 4 on the real number line. What is the directed distance from ⫺3 to 4? What is the directed distance from 4 to ⫺3? The distance between ⫺3 and 4 is given by

SOLUTION

ⱍ⫺3 ⫺ 4ⱍ ⫽ ⱍ⫺7ⱍ ⫽ 7

or

ⱍ4 ⫺ 共⫺3兲ⱍ ⫽ ⱍ7ⱍ ⫽ 7

ⱍa ⫺ bⱍ



or b ⫺ a



as shown in Figure 0.10. Distance = 7 −4 −3 −2 −1

0

1

2

3

4

5

x

FIGURE 0.10

The directed distance from ⫺3 to 4 is 4 ⫺ 共⫺3兲 ⫽ 7.

b⫺a

The directed distance from 4 to ⫺3 is ⫺3 ⫺ 4 ⫽ ⫺7.

a⫺b

✓CHECKPOINT 1 Determine the distance between ⫺2 and 6 on the real number line. What is the directed distance from ⫺2 to 6? What is the directed distance from 6 to ⫺2? ■

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10

CHAPTER 0

A Precalculus Review

Intervals Defined by Absolute Value Example 2

Defining an Interval on the Real Number Line

Find the interval on the real number line that contains all numbers that lie no more than two units from 3. Let x be any point in this interval. You need to find all x such that the distance between x and 3 is less than or equal to 2. This implies that

SOLUTION

ⱍx ⫺ 3ⱍ ≤ 2.

Requiring the absolute value of x ⫺ 3 to be less than or equal to 2 means that x ⫺ 3 must lie between ⫺2 and 2. So, you can write ⫺2 ≤ x ⫺ 3 ≤ 2. Solving this pair of inequalities, you have

⏐x − 3⏐ ≤ 2 2 units 2 units x

0

1

2

3

4

5

6

⫺2 ⫹ 3 ≤ x ⫺ 3 ⫹ 3 ≤ 2 ⫹ 3 1 ≤

x

≤ 5.

Solution set

So, the interval is 关1, 5 兴, as shown in Figure 0.11.

FIGURE 0.11

✓CHECKPOINT 2 Find the interval on the real number line that contains all numbers that lie no more than four units from 6. ■

Two Basic Types of Inequalities Involving Absolute Value

Let a and d be real numbers, where d > 0.

ⱍx ⫺ aⱍ ≤ d if and only if a ⫺ d ≤ x ≤ a ⫹ d. ⱍx ⫺ aⱍ ≥ d if and only if x ≤ a ⫺ d or a ⫹ d ≤ x.

Inequality STUDY TIP Be sure you see that inequalities of the form x ⫺ a ≥ d have solution sets consisting of two intervals. To describe the two intervals without using absolute values, you must use two separate inequalities, connected by an “or” to indicate union.



ⱍx ⫺ aⱍ ≤ d



ⱍx ⫺ aⱍ ≥ d

Larson Texts, Inc. • Final Pages • Applied Calculus 8e • CYAN

Interpretation

Graph

All numbers x whose distance from a is less than or equal to d.

d

x a−d

All numbers x whose distance from a is greater than or equal to d.

Short

d a

d

a+d

d x

a−d

4.0

a

a+d

Long

Copyright Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. MAGENTA 2009 YELLOW BLACK

SECTION 0.2

Absolute Value and Distance on the Real Number Line

11

Application Example 3 MAKE A DECISION

Quality Control

A large manufacturer hired a quality control firm to determine the reliability of a product. Using statistical methods, the firm determined that the manufacturer could expect 0.35% ± 0.17% of the units to be defective. If the manufacturer offers a money-back guarantee on this product, how much should be budgeted to cover the refunds on 100,000 units? (Assume that the retail price is $8.95.) Will the manufacturer have to establish a refund budget greater than $5000? Let r represent the percent of defective units (written in decimal form). You know that r will differ from 0.0035 by at most 0.0017. SOLUTION 0.0018

0.0052 r

0

0.002

0.004

0.006

520 x

0

100 200 300 400 500 600

(b) Number of defective units 1611

4654 C

0

Figure 0.12(a)

Now, letting x be the number of defective units out of 100,000, it follows that x ⫽ 100,000r and you have

(a) Percent of defective units 180

0.0035 ⫺ 0.0017 ≤ r ≤ 0.0035 ⫹ 0.0017 0.0018 ≤ r ≤ 0.0052

1000 2000 3000 4000 5000

(c) Cost of refunds

0.0018共100,000兲 ≤ 100,000r ≤ 0.0052共100,000兲 180 ≤ x ≤ 520. Figure 0.12(b) Finally, letting C be the cost of refunds, you have C ⫽ 8.95x. So, the total cost of refunds for 100,000 units should fall within the interval given by 180共8.95兲 ≤ 8.95x ≤ 520共8.95兲 $1611 ≤ C ≤ $4654.

Figure 0.12(c)

No, the refund budget will be less than $5000.

FIGURE 0.12

✓CHECKPOINT 3 Use the information in Example 3 to determine how much should be budgeted to cover refunds on 250,000 units. ■ Midpoint =

1611 + 4654 2

1611

= 3132.5 4654 C

0

1000 2000 3000 4000 5000

FIGURE 0.13

In Example 3, the manufacturer should expect to spend between $1611 and $4654 for refunds. Of course, the safer budget figure for refunds would be the higher of these estimates. However, from a statistical point of view, the most representative estimate would be the average of these two extremes. Graphically, the average of two numbers is the midpoint of the interval with the two numbers as endpoints, as shown in Figure 0.13. Midpoint of an Interval

The midpoint of the interval with endpoints a and b is found by taking the average of the endpoints. Midpoint ⫽

a⫹b 2

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12

CHAPTER 0

A Precalculus Review

Exercises 0.2

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 6, find (a) the directed distance from a to b, (b) the directed distance from b to a, and (c) the distance between a and b. 1. a ⫽ 126, b ⫽ 75

2. a ⫽ ⫺126, b ⫽ ⫺75

3. a ⫽ 9.34, b ⫽ ⫺5.65

4. a ⫽ ⫺2.05, b ⫽ 4.25

5. a ⫽

16 5,

b ⫽ 112 75

18 6. a ⫽ ⫺ 5 , b ⫽

61 15

In Exercises 7–18, use absolute values to describe the given interval (or pair of intervals) on the real number line. 7. 关⫺2, 2兴 9. 共⫺ ⬁, ⫺2兲 傼 共2, ⬁兲

8. 共⫺3, 3兲 10. 共⫺ ⬁, ⫺3兴 傼 关3, ⬁兲

11. 关2, 8兴

12. 共⫺7, ⫺1兲

13. 共⫺ ⬁, 0兲 傼 共4, ⬁兲

14. 共⫺ ⬁, 20兲 傼 共24, ⬁兲

15. All numbers less than three units from 5 16. All numbers more than five units from 2 17. y is at most two units from a.

In Exercises 19–34, solve the inequality and sketch the graph of the solution on the real number line.

ⱍ ⱍ ⱍ3xⱍ > 12 ⱍ3x ⫹ 1ⱍ ≥ 4 ⱍ2x ⫹ 1ⱍ < 5 ⱍ25 ⫺ xⱍ ≥ 20

19. x < 4

20. 2x < 6

x > 3 21. 2

22.

23. x ⫺ 5 < 2

24.

x⫺3 ≥ 5 2

26.

25.

ⱍⱍ ⱍ ⱍ ⱍ ⱍ

ⱍ ⱍ ⱍ9 ⫺ 2xⱍ < 1 ⱍx ⫺ aⱍ ≤ b, b > 0

27. 10 ⫺ x > 4

28.

29.

30. 1 ⫺

31.

ⱍ ⱍ

3x ⫺ a < 2b, b > 0 33. 4

43. Heights of a Population The heights h of two-thirds of the members of a population satisfy the inequality





h ⫺ 68.5 ≤ 1 2.7

where h is measured in inches. Determine the interval on the real number line in which these heights lie. 44. Biology The American Kennel Club has developed guidelines for judging the features of various breeds of dogs. For collies, the guidelines specify that the weights for males satisfy the inequality





w ⫺ 67.5 ≤ 1 7.5

where w is measured in pounds. Determine the interval on the real number line in which these weights lie. 45. Production The estimated daily production x at a refinery is given by

18. y is less than h units from c.

ⱍⱍ

42. Stock Price A stock market analyst predicts that over the next year the price p of a stock will not change from its current price of $33.15 by more than $2. Use absolute values to write this prediction as an inequality.

ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ

2x < 1 3

32. 2x ⫺ a ≥ b, b > 0 34. a ⫺

5x > b, b > 0 2

ⱍx ⫺ 200,000ⱍ ≤ 25,000

where x is measured in barrels of oil. Determine the high and low production levels. 46. Manufacturing The acceptable weights for a 20-ounce cereal box are given by x ⫺ 20 ≤ 0.75, where x is measured in ounces. Determine the high and low weights for the cereal box.





Budget Variance In Exercises 47–50, (a) use absolute value notation to represent the two intervals in which expenses must lie if they are to be within $500 and within 5% of the specified budget amount and (b) using the more stringent constraint, determine whether the given expense is at variance with the budget restriction. Budget

Expense

In Exercises 35– 40, find the midpoint of the given interval.

47. Utilities

$4750.00

$5116.37

48. Insurance

$15,000.00

$14,695.00

35. 关8, 24兴

36. 关7.3, 12.7兴

49. Maintenance

$20,000.00

$22,718.35

37. 关⫺6.85, 9.35兴

38. 关⫺4.6, ⫺1.3兴

50. Taxes

$7500.00

$8691.00

39.



⫺ 12, 34



40.

关 兴 5 5 6, 2

41. Chemistry Copper has a melting point M within 0.2°C of 1083.4°C. Use absolute values to write the range as an inequality.

Item

51. Quality Control In determining the reliability of a product, a manufacturer determines that it should expect 0.05% ± 0.01% of the units to be defective. If the manufacturer offers a money-back guarantee on this product, how much should be budgeted to cover the refunds on 150,000 units? (Assume that the retail price is $195.99.)

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 0.3

Exponents and Radicals

Section 0.3 ■ Evaluate expressions involving exponents or radicals.

Exponents and Radicals

■ Simplify expressions with exponents. ■ Find the domains of algebraic expressions.

Expressions Involving Exponents or Radicals Properties of Exponents

⭈x.

. .x

1. Whole-number exponents:

xn ⫽ x ⭈ x

2. Zero exponent:

x ⫽ 1, x ⫽ 0

3. Negative exponents:

x⫺n ⫽

4. Radicals (principal nth root):

n x ⫽ a 冪

5. Rational exponents 共1兾n兲:

n x x 1兾n ⫽ 冪

6. Rational exponents 共m兾n兲:

n x x m兾n ⫽ 共x1兾n兲m ⫽ 共冪 兲

n factors

STUDY TIP If n is even, then the principal nth root is positive. For example, 4 81 ⫽ ⫹3. 冪4 ⫽ ⫹2 and 冪



Evaluating Expressions

a. y ⫽ ⫺2x 2

x⫽4

y ⫽ ⫺2共4 2兲 ⫽ ⫺2共16兲 ⫽ ⫺32

b. y ⫽ 3x⫺3

x ⫽ ⫺1

y ⫽ 3共⫺1兲⫺3 ⫽

c. y ⫽ 共⫺x兲 2

x⫽

d. y ⫽

a. y ⫽ ■

2 x ⫽ 冪x 冪

Substitution

2 x⫺2

Expression

Evaluate y ⫽ 4x1兾3 for x ⫽ 8.

x ⫽ an

x-Value

Example 2

✓CHECKPOINT 2

x⫽0

m

7. Special convention (square root):

Expression

Evaluate y ⫽ 4x⫺2 for x ⫽ 3.

1 , xn

n xm x m兾n ⫽ 共x m兲1兾n ⫽ 冪

Example 1

✓CHECKPOINT 1

0

2x 1兾2

3 x2 b. y ⫽ 冪

1 2

x⫽3

冢 12冣

y⫽ ⫺ y⫽

2



3 3 ⫽ ⫽ ⫺3 3 共⫺1兲 ⫺1

1 4

2 ⫽ 2共32兲 ⫽ 18 3⫺2

Evaluating Expressions x-Value

Substitution

x⫽4

y ⫽ 2冪4 ⫽ 2共2兲 ⫽ 4

x⫽8

y ⫽ 8 2兾3 ⫽ 共81兾3兲 2 ⫽ 22 ⫽ 4

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13

14

CHAPTER 0

A Precalculus Review

Operations with Exponents TECHNOLOGY

Operations with Exponents

Graphing utilities perform the established order of operations when evaluating an expression. To see this, try entering the expressions



1200 1 ⫹

0.09 12



1. Multiplying like bases:

x n x m ⫽ x n⫹m

Add exponents.

2. Dividing like bases:

xn ⫽ x n⫺m xm

Subtract exponents.

3. Removing parentheses:

12 ⭈ 6

共xy兲n ⫽ x n y n x n xn ⫽ n y y 共x n兲m ⫽ x nm

冢冣

and 1200



1⫹

冢0.09 12 冣

4. Special conventions:

12 ⭈ 6

⫺x n ⫽ ⫺ 共x n兲, ⫺x n ⫽ 共⫺x兲n cx n ⫽ c共x n兲, cx n ⫽ 共cx兲n m m m x n ⫽ x共n 兲, x n ⫽ 共x n兲m

into your graphing utility to see that the expressions result in different values.*

Example 3

Simplifying Expressions with Exponents

Simplify each expression. a. 2x 2共x 3兲

3 x b. 共3x兲 2冪

c.

3x2 共x 1兾2兲3

5x4 共x2兲3

e. x⫺1共2x 2兲

f.

⫺ 冪x 5x⫺1

d.

SOLUTION

a. 2x 2共x 3兲 ⫽ 2x 2⫹3 ⫽ 2x 5

x n x m ⫽ x n⫹m

3 x ⫽ 9x 2x 1兾3 ⫽ 9x 2⫹ 共1兾3兲 ⫽ 9x 7兾3 b. 共3x兲2冪

x n x m ⫽ x n⫹m

c.

3x 2 x2 ⫽ 3 ⫽ 3x 2⫺ 共3兾2兲 ⫽ 3x 1兾2 共x 1兾2兲 3 x 3兾2

共x n兲 m ⫽ x nm,

xn ⫽ x n⫺m xm

d.

5x 4 5x 4 5 ⫽ 6 ⫽ 5x 4⫺6 ⫽ 5x⫺2 ⫽ 2 2 3 共x 兲 x x

共x n兲 m ⫽ x nm,

xn ⫽ x n⫺m xm

冢 冣

e. x⫺1共2x 2兲 ⫽ 2x⫺1x 2 ⫽ 2x 2⫺1 ⫽ 2x f.

x n x m ⫽ x n⫹m

⫺ 冪x 1 x1兾2 1 1 ⫽ ⫺ ⫽ ⫺ x 共1兾2兲 ⫹1 ⫽ ⫺ x 3兾2 ⫺1 ⫺1 5x 5 x 5 5

冢 冣

xn ⫽ x n⫺m xm

✓CHECKPOINT 3 Simplify each expression. a. 3x2 共x 4兲

b. 共2x兲3冪x

c.

4x2 共x1兾3兲2



*Specific calculator keystroke instructions for operations in this and other technology boxes can be found at college.hmco.com/info/larsonapplied.

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SECTION 0.3

Exponents and Radicals

15

Note in Example 3 that one characteristic of simplified expressions is the absence of negative exponents. Another characteristic of simplified expressions is that sums and differences are written in factored form. To do this, you can use the Distributive Property. abx n ⫹ acx n⫹m ⫽ ax n共b ⫹ cx m兲 Study the next example carefully to be sure that you understand the concepts involved in the factoring process.

Example 4

Simplifying by Factoring

Simplify each expression by factoring. a. 2x 2 ⫺ x 3

✓CHECKPOINT 4 Simplify each expression by factoring. a. x3 ⫺ 2x b. 2x1兾2 ⫹ 8x3兾2



STUDY TIP To check that the simplified expression is equivalent to the original expression, try substituting values for x into each expression.

b. 2x 3 ⫹ x 2

c. 2x1兾2 ⫹ 4x 5兾2

d. 2x⫺1兾2 ⫹ 3x 5兾2

SOLUTION

a. 2x 2 ⫺ x 3 ⫽ x 2共2 ⫺ x兲 b. 2x 3 ⫹ x 2 ⫽ x 2共2x ⫹ 1兲 c. 2x 1兾2 ⫹ 4x 5兾2 ⫽ 2x 1兾2共1 ⫹ 2x 2兲 d. 2x⫺1兾2 ⫹ 3x 5兾2 ⫽ x⫺1兾2共2 ⫹ 3x 3兲 ⫽

2 ⫹ 3x 3 冪x

Many algebraic expressions obtained in calculus occur in unsimplified form. For instance, the two expressions shown in the following example are the result of an operation in calculus called differentiation. 关The first is the derivative of 2共x ⫹ 1兲3兾2共2x ⫺ 3兲5兾2, and the second is the derivative of 2共x ⫹ 1兲1兾2共2x ⫺ 3兲5兾2.兴

Example 5

Simplifying by Factoring

Simplify each expression by factoring. a. 3共x ⫹ 1兲1兾2共2x ⫺ 3兲5兾2 ⫹ 10共x ⫹ 1兲3兾2共2x ⫺ 3兲3兾2 ⫽ 共x ⫹ 1兲1兾2共2x ⫺ 3兲 3兾2关3共2x ⫺ 3兲 ⫹ 10共x ⫹ 1兲兴 ⫽ 共x ⫹ 1兲1兾2共2x ⫺ 3兲 3兾2共6x ⫺ 9 ⫹ 10x ⫹ 10兲 ⫽ 共x ⫹ 1兲 1兾2共2x ⫺ 3兲 3兾2共16x ⫹ 1兲 b. 共x ⫹ 1兲⫺1兾2共2x ⫺ 3兲5兾2 ⫹ 10共x ⫹ 1兲1兾2共2x ⫺ 3兲 3兾2

✓CHECKPOINT 5 Simplify the expression by factoring. 共x ⫹ 2兲1兾2共3x ⫺ 1兲3兾2 ⫹ 4共x ⫹ 2兲⫺1兾2共3x ⫺ 1兲5兾2 ■

⫽ 共x ⫹ 1兲⫺1兾2共2x ⫺ 3兲 3兾2关共2x ⫺ 3兲 ⫹ 10共x ⫹ 1兲兴 ⫽ 共x ⫹ 1兲⫺1兾2共2x ⫺ 3兲 3兾2共2x ⫺ 3 ⫹ 10x ⫹ 10兲 ⫽ 共x ⫹ 1兲⫺1兾2共2x ⫺ 3兲 3兾2共12x ⫹ 7兲 ⫽

共2x ⫺ 3兲 3兾2共12x ⫹ 7兲 共x ⫹ 1兲1兾2

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16

CHAPTER 0

A Precalculus Review

Example 6 shows some additional types of expressions that can occur in calculus. 关The expression in Example 6(d) is an antiderivative of 共x ⫹ 1兲2兾3共2x ⫹ 3兲, and the expression in Example 6(e) is the derivative of 共x ⫹ 2兲 3兾共x ⫺ 1兲 3.兴 TECHNOLOGY

>

A graphing utility offers several ways to calculate rational exponents and radicals. You should be familiar with the x-squared key x 2 . This key squares the value of an expression. For rational exponents or exponents other than 2, use the key. For radical expressions, you can use the square root key 冪 , the cube root key 冪3 , or the xth root key 冪x . Consult your graphing utility user’s guide for specific keystrokes you can use to evaluate rational exponents and radical expressions. Use a graphing utility to evaluate each expression. a. 共⫺8兲2兾3

b. 共16 ⫺ 5兲 4

c. 冪576

3 729 d. 冪

Example 6

Simplify each expression by factoring. a. b.



x

3 3 共x ⫹ 1兲 5兾3 ⫹ 共x ⫹ 1兲 8兾3 5 4

e.

3共x ⫹ 2兲 2共x ⫺ 1兲 3 ⫺ 3共x ⫹ 2兲 3共x ⫺ 1兲 2 关共x ⫺ 1兲 3兴 2

SOLUTION

a. b.

3x 2 ⫹ x 4 x 2共3 ⫹ x 2兲 x 2⫺1共3 ⫹ x 2兲 x共3 ⫹ x 2兲 ⫽ ⫽ ⫽ 2x 2x 2 2 冪x ⫹ x 3兾2

x



1⫹x x1兾2共1 ⫹ x兲 1⫹x ⫽ 1⫺ 共1兾2兲 ⫽ x x 冪x

c. 共9x ⫹ 2兲⫺1兾3 ⫹ 18共9x ⫹ 2兲 ⫽ 共9x ⫹ 2兲⫺1兾3 关1 ⫹ 18共9x ⫹ 2兲4兾3兴 ⫽

e.

5x3 ⫹ x6 3x

冪x ⫹ x 3兾2

d.

d.

Simplify the expression by factoring.

3x 2 ⫹ x 4 2x

c. 共9x ⫹ 2兲⫺1兾3 ⫹ 18共9x ⫹ 2兲

4 共16兲 3 e. 冪

✓CHECKPOINT 6

Factors Involving Quotients

1 ⫹ 18共9x ⫹ 2兲4兾3 3 冪 9x ⫹ 2

3 3 12 15 共x ⫹ 1兲 5兾3 ⫹ 共x ⫹ 1兲 8兾3 ⫽ 共x ⫹ 1兲 5兾3 ⫹ 共x ⫹ 1兲 8兾3 5 4 20 20 ⫽

3 共x ⫹ 1兲 5兾3关4 ⫹ 5共x ⫹ 1兲兴 20



3 共x ⫹ 1兲 5兾3共4 ⫹ 5x ⫹ 5兲 20



3 共x ⫹ 1兲 5兾3共5x ⫹ 9兲 20

3共x ⫹ 2兲 2共x ⫺ 1兲 3 ⫺ 3共x ⫹ 2兲 3共x ⫺ 1兲 2 关共x ⫺ 1兲 3兴 2 3共x ⫹ 2兲 2共x ⫺ 1兲 2 关共x ⫺ 1兲 ⫺ 共x ⫹ 2兲兴 ⫽ 共x ⫺ 1兲 6 3共x ⫹ 2兲2共x ⫺ 1 ⫺ x ⫺ 2兲 ⫽ 共x ⫺ 1兲6⫺2 ⫺9共x ⫹ 2兲 2 ⫽ 共x ⫺ 1兲 4

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 0.3

Exponents and Radicals

17

Domain of an Algebraic Expression When working with algebraic expressions involving x, you face the potential difficulty of substituting a value of x for which the expression is not defined (does not produce a real number). For example, the expression 冪2x ⫹ 3 is not defined when x ⫽ ⫺2 because 冪2共⫺2兲 ⫹ 3 is not a real number. The set of all values for which an expression is defined is called its domain. So, the domain of 冪2x ⫹ 3 is the set of all values of x such that 冪2x ⫹ 3 is a real number. In order for 冪2x ⫹ 3 to represent a real number, it is necessary that 2x ⫹ 3 ≥ 0. In other words, 冪2x ⫹ 3 is defined only for those values of x that 3 lie in the interval 关⫺ 2, ⬁兲, as shown in Figure 0.14. 2x + 3 is not defined for these x. − 32

2x + 3 is defined for these x. x

−3

−2

−1

0

1

2

3

FIGURE 0.14

Example 7

Finding the Domain of an Expression

Find the domain of each expression. a. 冪3x ⫺ 2 b.

1 冪3x ⫺ 2

3 9x ⫹ 1 c. 冪

SOLUTION

a. The domain of 冪3x ⫺ 2 consists of all x such that 3x ⫺ 2 ≥ 0

Expression must be nonnegative.

which implies that x ≥ 23. So, the domain is 关23, ⬁兲. b. The domain of 1兾冪3x ⫺ 2 is the same as the domain of 冪3x ⫺ 2, except that 1兾冪3x ⫺ 2 is not defined when 3x ⫺ 2 ⫽ 0. Because this occurs when x ⫽ 23, the domain is 共23, ⬁兲. 3 9x ⫹ 1 is defined for all real numbers, its domain is 共⫺ c. Because 冪 ⬁, ⬁兲.

✓CHECKPOINT 7 Find the domain of each expression. a. 冪x ⫺ 2 b.

1 冪x ⫺ 2

3 x ⫺ 2 c. 冪



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

18

CHAPTER 0

A Precalculus Review

Exercises 0.3

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–20, evaluate the expression for the given value of x. Expression

x-Value

43.

共x ⫹ 1兲共x ⫺ 1兲2 ⫺ 共x ⫺ 1兲3 共x ⫹ 1兲2

Expression

x-Value

44. 共x 4 ⫹ 2兲3共x ⫹ 3兲⫺1兾2 ⫹ 4x 3共x 4 ⫹ 2兲2共x ⫹ 3兲1兾2 In Exercises 45–52, find the domain of the given expression.

1. ⫺2x 3

x⫽3

2.

x2 3

x⫽6

3. 4x⫺3

x⫽2

4. 7x⫺2

x⫽5

45. 冪x ⫺ 4

46. 冪5 ⫺ 2x

x⫽3

6. x ⫺ 4x⫺2

x⫽3

47. 冪x 2 ⫹ 3

48. 冪4x 2 ⫹ 1

x ⫽ ⫺2

8. 5共⫺x兲 3

x⫽3

49.

1 共⫺x兲⫺3

x⫽4

5.

x⫺1

1⫹ x⫺1

7. 3x 2 ⫺ 4x 3

9. 6x 0 ⫺ 共6x兲0 x ⫽ 10 11.

3 x2 冪

13.

x⫺1兾2

10.

1 9

x ⫽ 27

12.

冪x 3

x⫽4

14.

x⫺3兾4

15. x⫺2兾5

x ⫽ ⫺32

16. 共x2兾3兲3

17. 500x 60

x ⫽ 1.01

18.

10,000 x 120

x ⫽ 1.075

20.

6 x 冪

x ⫽ 325

19.

3 x 冪

x ⫽ ⫺154

x⫽

x ⫽ 16 x ⫽ 10

In Exercises 21–30, simplify the expression. 21. 6y⫺2 共2y 4兲⫺3

22. z⫺3共3z 4兲

23. 10共x 2兲 2

24. 共4x 3兲 2

7x 2 25. ⫺3 x

x ⫺3 26. 冪x

27.

10共x ⫹ y兲3 4共x ⫹ y兲⫺2

3x冪x 29. 1兾2 x

2 3

28.

冢12s9s 冣

30.



3 x2 冪



3

In Exercises 31–36, simplify by removing all possible factors from the radical. 3 16 冪 27

31. 冪8

32.

3 54x 5 33. 冪

4 共3x 2 y 3兲 4 34. 冪

3 144x 9 y⫺4 z 5 35. 冪

4 32xy 5z⫺8 36. 冪

In Exercises 37– 44, simplify each expression by factoring. 37. 3x 3 ⫺ 12x

38. 8x 4 ⫺ 6x 2

39. 2x 5兾2 ⫹ x⫺1兾2

40. 5x 3兾2 ⫺ x⫺3兾2

41. 3x共x ⫹ 1兲3兾2 ⫺ 6共x ⫹ 1兲1兾2 42. 2x 共x ⫺ 1兲5兾2 ⫺ 4共x ⫺ 1兲3兾2

51.

1

50.

3 x ⫺ 4 冪

冪x ⫹ 2

52.

1⫺x

1 3 x ⫹ 4 冪

1 冪2x ⫹ 3

⫹ 冪6 ⫺ 4x

Compound Interest In Exercises 53–56, a certificate of deposit has a principal of P and an annual percentage rate of r (expressed as a decimal) compounded n times per year. Enter the compound interest formula



AⴝP 11

r n



N

into a graphing utility and use it to find the balance after N compoundings. 53. P ⫽ $10,000, r ⫽ 6.5%, n ⫽ 12,

N ⫽ 120

54. P ⫽ $7000, r ⫽ 5%, n ⫽ 365,

N ⫽ 1000

55. P ⫽ $5000, r ⫽ 5.5%, n ⫽ 4,

N ⫽ 60

56. P ⫽ $8000, r ⫽ 7%, n ⫽ 6, 57. Period of a Pendulum T ⫽ 2␲

N ⫽ 90

The period of a pendulum is

冪32L

where T is the period in seconds and L is the length of the pendulum in feet. Find the period of a pendulum whose length is 4 feet. 58. Annuity A balance A, after n annual payments of P dollars have been made into an annuity earning an annual percentage rate of r compounded annually, is given by A ⫽ P共1 ⫹ r兲 ⫹ P共1 ⫹ r兲 2 ⫹ . . . ⫹ P共1 ⫹ r兲 n. Rewrite this formula by completing the following factor兲. ization: A ⫽ P共1 ⫹ r兲共 59. Extended Application To work an extended application analyzing the population per square mile of the United States, visit this text’s website at college.hmco.com (Data Source: U.S. Census Bureau)

The symbol indicates when to use graphing technology or a symbolic computer algebra system to solve a problem or an exercise. The solutions to other problems or exercises may also be facilitated by use of appropriate technology.

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SECTION 0.4

Factoring Polynomials

19

Section 0.4 ■ Use special products and factorization techniques to factor polynomials.

Factoring Polynomials

■ Find the domains of radical expressions. ■ Use synthetic division to factor polynomials of degree three or more. ■ Use the Rational Zero Theorem to find the real zeros of polynomials.

Factorization Techniques The Fundamental Theorem of Algebra states that every nth-degree polynomial an x n ⫹ an⫺1 x n⫺1 ⫹ . . . ⫹ a1 x ⫹ a0, an ⫽ 0 has precisely n zeros. (The zeros may be repeated or imaginary.) The problem of finding the zeros of a polynomial is equivalent to the problem of factoring the polynomial into linear factors. Special Products and Factorization Techniques

Quadratic Formula

Example ⫺b ± 冪b 2 ⫺ 4ac x⫽ 2a

ax 2 ⫹ bx ⫹ c ⫽ 0 Special Products x2

a2

⫺ ⫽ 共x 3 x ⫺ a 3 ⫽ 共x x 3 ⫹ a 3 ⫽ 共x x 4 ⫺ a 4 ⫽ 共x

⫺ ⫺ ⫹ ⫺

x⫽

⫺3 ± 冪13 2

Examples a兲共x ⫹ a兲 a兲共x 2 ⫹ ax ⫹ a 2兲 a兲共x 2 ⫺ ax ⫹ a 2兲 a兲共x ⫹ a兲共x 2 ⫹ a 2兲

Binomial Theorem

x2 x3 x3 x4

⫺ ⫺ ⫹ ⫺

9 ⫽ 共x ⫺ 3兲共x ⫹ 3兲 8 ⫽ 共x ⫺ 2兲共x 2 ⫹ 2x ⫹ 4兲 64 ⫽ 共x ⫹ 4兲共x 2 ⫺ 4x ⫹ 16兲 16 ⫽ 共x ⫺ 2兲共x ⫹ 2兲共x 2 ⫹ 4兲

Examples

共x ⫹ a兲 ⫽ ⫹ 2ax ⫹ 共x ⫹ 3兲 2 ⫽ x 2 ⫹ 6x ⫹ 9 共x ⫺ a兲 2 ⫽ x 2 ⫺ 2ax ⫹ a 2 共x 2 ⫺ 5兲 2 ⫽ x 4 ⫺ 10x 2 ⫹ 25 共x ⫹ a兲 3 ⫽ x 3 ⫹ 3ax 2 ⫹ 3a 2x ⫹ a 3 共x ⫹ 2兲 3 ⫽ x 3 ⫹ 6x 2 ⫹ 12x ⫹ 8 3 3 2 2 3 共x ⫺ a兲 ⫽ x ⫺ 3ax ⫹ 3a x ⫺ a 共x ⫺ 1兲 3 ⫽ x 3 ⫺ 3x 2 ⫹ 3x ⫺ 1 共x ⫹ a兲 4 ⫽ x 4 ⫹ 4ax 3 ⫹ 6a 2 x 2 ⫹ 4a 3x ⫹ a 4 共x ⫹ 2兲 4 ⫽ x 4 ⫹ 8x 3 ⫹ 24x 2 ⫹ 32x ⫹ 16 共x ⫺ a兲 4 ⫽ x 4 ⫺ 4ax 3 ⫹ 6a 2x 2 ⫺ 4a 3 x ⫹ a 4 共x ⫺ 4兲 4 ⫽ x 4 ⫺ 16x 3 ⫹ 96x 2 ⫺ 256x ⫹ 256 n共n ⫺ 1兲 2 n⫺2 n共n ⫺ 1兲共n ⫺ 2兲 3 n⫺3 . . . 共x ⫹ a兲n ⫽ x n ⫹ nax n⫺1 ⫹ ⫹ ⫹ ⫹ na n⫺1 x ⫹ a n * a x ax 2! 3! n共n ⫺ 1兲 2 n⫺2 n共n ⫺ 1兲共n ⫺ 2兲 3 n⫺3 . . . ⫺ ⫹ ± na n⫺1x ⫿ a n a x ax 共x ⫺ a兲 n ⫽ x n ⫺ nax n⫺1 ⫹ 2! 3! 2

x2

x 2 ⫹ 3x ⫺ 1 ⫽ 0

a2

Factoring by Grouping acx 3 ⫹ adx 2 ⫹ bcx ⫹ bd ⫽ ax 2共cx ⫹ d兲 ⫹ b共cx ⫹ d兲 ⫽ 共ax 2 ⫹ b兲共cx ⫹ d兲

Example 3x 3 ⫺ 2x 2 ⫺ 6x ⫹ 4 ⫽ x 2共3x ⫺ 2兲 ⫺ 2共3x ⫺ 2兲 ⫽ 共x 2 ⫺ 2兲共3x ⫺ 2兲

* The factorial symbol ! is defined as follows: 0! ⫽ 1, 1! ⫽ 1, 2! ⫽ 2 ⭈ 1 ⫽ 2, 3! ⫽ 3 ⭈ 2 ⭈ 1 ⫽ 6, 4! ⫽ 4 ⭈ 3 ⭈ 2 ⭈ 1 ⫽ 24, and so on.

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20

CHAPTER 0

A Precalculus Review

Example 1

Applying the Quadratic Formula

Use the Quadratic Formula to find all real zeros of each polynomial. a. 4x 2 ⫹ 6x ⫹ 1

b. x 2 ⫹ 6x ⫹ 9

c. 2x 2 ⫺ 6x ⫹ 5

SOLUTION

a. Using a ⫽ 4, b ⫽ 6, and c ⫽ 1, you can write x⫽

⫺b ± 冪b 2 ⫺ 4ac ⫺6 ± 冪36 ⫺ 16 ⫽ 2a 8 ⫽

⫺6 ± 冪20 8



⫺6 ± 2冪5 8



2共⫺3 ± 冪5 兲 2共4兲



⫺3 ± 冪5 . 4

So, there are two real zeros: x⫽ STUDY TIP Try solving Example 1(b) by factoring. Do you obtain the same answer?

⫺3 ⫺ 冪5 ⬇ ⫺1.309 4

and

x⫽

⫺3 ⫹ 冪5 ⬇ ⫺0.191. 4

b. In this case, a ⫽ 1, b ⫽ 6, and c ⫽ 9, and the Quadratic Formula yields x⫽

⫺b ± 冪b 2 ⫺ 4ac ⫺6 ± 冪36 ⫺ 36 6 ⫽ ⫽ ⫺ ⫽ ⫺3. 2a 2 2

So, there is one (repeated) real zero: x ⫽ ⫺3. c. For this quadratic equation, a ⫽ 2, b ⫽ ⫺6, and c ⫽ 5. So, x⫽

⫺b ± 冪b 2 ⫺ 4ac 6 ± 冪36 ⫺ 40 6 ± 冪⫺4 . ⫽ ⫽ 2a 4 4

Because 冪⫺4 is imaginary, there are no real zeros.

✓CHECKPOINT 1 Use the Quadratic Formula to find all real zeros of each polynomial. a. 2x2 ⫹ 4x ⫹ 1

b. x2 ⫺ 8x ⫹ 16

c. 2x2 ⫺ x ⫹ 5



The zeros in Example 1(a) are irrational, and the zeros in Example 1(c) are imaginary. In both of these cases the quadratic is said to be irreducible because it cannot be factored into linear factors with rational coefficients. The next example shows how to find the zeros associated with reducible quadratics. In this example, factoring is used to find the zeros of each quadratic. Try using the Quadratic Formula to obtain the same zeros.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 0.4

Example 2

Factoring Polynomials

21

Factoring Quadratics

Find the zeros of each quadratic polynomial. a. x 2 ⫺ 5x ⫹ 6

b. x 2 ⫺ 6x ⫹ 9

c. 2x 2 ⫹ 5x ⫺ 3

SOLUTION

a. Because x 2 ⫺ 5x ⫹ 6 ⫽ 共x ⫺ 2兲共x ⫺ 3兲 STUDY TIP The zeros of a polynomial in x are the values of x that make the polynomial zero. To find the zeros, factor the polynomial into linear factors and set each factor equal to zero. For instance, the zeros of 共x ⫺ 2兲共x ⫺ 3兲 occur when x ⫺ 2 ⫽ 0 and x ⫺ 3 ⫽ 0.

the zeros are x ⫽ 2 and x ⫽ 3. b. Because x 2 ⫺ 6x ⫹ 9 ⫽ 共x ⫺ 3兲2 the only zero is x ⫽ 3. c. Because 2x 2 ⫹ 5x ⫺ 3 ⫽ 共2x ⫺ 1兲共x ⫹ 3兲 the zeros are x ⫽ 12 and x ⫽ ⫺3.

✓CHECKPOINT 2 Find the zeros of each quadratic polynomial. a. x2 ⫺ 2x ⫺ 15

Example 3 Values of 冪x2 ⴚ 3x 1 2

x

冪x 2 ⫺ 3x ⫹ 2

0

冪2

1

0

1.5

Undefined

2

0

3

冪2

b. x2 ⫹ 2x ⫹ 1

Find the domain of 冪x 2 ⫹ x ⫺ 2. ■



Finding the Domain of a Radical Expression

Find the domain of 冪x 2 ⫺ 3x ⫹ 2. SOLUTION

Because

x 2 ⫺ 3x ⫹ 2 ⫽ 共x ⫺ 1兲共x ⫺ 2兲 you know that the zeros of the quadratic are x ⫽ 1 and x ⫽ 2. So, you need to test the sign of the quadratic in the three intervals 共⫺ ⬁, 1兲, 共1, 2兲, and 共2, ⬁兲, as shown in Figure 0.15. After testing each of these intervals, you can see that the quadratic is negative in the center interval and positive in the outer two intervals. Moreover, because the quadratic is zero when x ⫽ 1 and x ⫽ 2, you can conclude that the domain of 冪x 2 ⫺ 3x ⫹ 2 is

共⫺ ⬁, 1兴 傼 关2, ⬁兲.

✓CHECKPOINT 3

c. 2x2 ⫺ 7x ⫹ 6

x 2 − 3x + 2 is defined.

Domain

x 2 − 3x + 2 is not defined.

x 2 − 3x + 2 is defined. x

−1

0

1

2

3

4

FIGURE 0.15

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22

CHAPTER 0

A Precalculus Review

Factoring Polynomials of Degree Three or More It can be difficult to find the zeros of polynomials of degree three or more. However, if one of the zeros of a polynomial is known, then you can use that zero to reduce the degree of the polynomial. For example, if you know that x ⫽ 2 is a zero of x 3 ⫺ 4x 2 ⫹ 5x ⫺ 2, then you know that 共x ⫺ 2兲 is a factor, and you can use long division to factor the polynomial as shown. x 3 ⫺ 4x 2 ⫹ 5x ⫺ 2 ⫽ 共x ⫺ 2兲共x 2 ⫺ 2x ⫹ 1兲 ⫽ 共x ⫺ 2兲共x ⫺ 1兲共x ⫺ 1兲 As an alternative to long division, many people prefer to use synthetic division to reduce the degree of a polynomial. Synthetic Division for a Cubic Polynomial

Given: x ⫽ x1 is a zero of ax 3 ⫹ bx 2 ⫹ cx ⫹ d. x1

a

b

c

a

d

Vertical pattern: Add terms.

0

Diagonal pattern: Multiply by x1.

Coefficients for quadratic factor

Performing synthetic division on the polynomial x 3 ⫺ 4x 2 ⫹ 5x ⫺ 2 using the given zero, x ⫽ 2, produces the following. 2

共x ⫺ 2兲共

1

⫺4 2

5 ⫺4

⫺2 2

1

⫺2

1

0

x2

⫺ 2x ⫹ 1兲 ⫽

x3

⫺ 4x 2 ⫹ 5x ⫺ 2

When you use synthetic division, remember to take all coefficients into account—even if some of them are zero. For instance, if you know that x ⫽ ⫺2 is a zero of x 3 ⫹ 3x ⫹ 14, you can apply synthetic division as shown. ⫺2

x2

共x ⫹ 2兲共

1

0 ⫺2

1

⫺2

⫺ 2x ⫹ 7兲 ⫽

3 14 4 ⫺14 7 x3

0

⫹ 3x ⫹ 14

STUDY TIP The algorithm for synthetic division given above works only for divisors of the form x ⫺ x 1. Remember that x ⫹ x1 ⫽ x ⫺ 共⫺x1 兲.

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SECTION 0.4

Factoring Polynomials

23

The Rational Zero Theorem There is a systematic way to find the rational zeros of a polynomial. You can use the Rational Zero Theorem (also called the Rational Root Theorem). Rational Zero Theorem

If a polynomial an x n ⫹ a n⫺1 x n⫺1 ⫹ . . . ⫹ a1 x ⫹ a 0 has integer coefficients, then every rational zero is of the form x ⫽ p兾q, where p is a factor of a 0, and q is a factor of a n.

Example 4

Using the Rational Zero Theorem

Find all real zeros of the polynomial. 2x 3 ⫹ 3x 2 ⫺ 8x ⫹ 3 SOLUTION

2 x 3 ⫹ 3x 2 ⫺ 8x ⫹ 3 Factors of constant term: ± 1, ± 3 STUDY TIP In Example 4, you can check that the zeros are correct by substituting into the original polynomial. Check that x ⫽ 1 is a zero. 2共1兲3 ⫹ 3共1兲2 ⫺ 8共1兲 ⫹ 3 ⫽2⫹3⫺8⫹3 ⫽0 Check that x ⫽ 12 is a zero. 1 3 1 2 1 ⫹3 ⫺8 ⫹3 2 2 2 2 1 3 ⫽ ⫹ ⫺4⫹3 4 4 ⫽0

冢冣

冢冣

冢冣

Check that x ⫽ ⫺3 is a zero. 2共⫺3兲3 ⫹ 3共⫺3兲 2 ⫺ 8共⫺3兲 ⫹ 3 ⫽ ⫺54 ⫹ 27 ⫹ 24 ⫹ 3 ⫽0

Factors of leading coefficient: ± 1, ± 2 The possible rational zeros are the factors of the constant term divided by the factors of the leading coefficient. 1 1 3 3 1, ⫺1, 3, ⫺3, , ⫺ , , ⫺ 2 2 2 2 By testing these possible zeros, you can see that x ⫽ 1 works. 2共1兲3 ⫹ 3共1兲 2 ⫺ 8共1兲 ⫹ 3 ⫽ 2 ⫹ 3 ⫺ 8 ⫹ 3 ⫽ 0 Now, by synthetic division you have the following. 1

2

3 2

⫺8 5

3 ⫺3

2

5

⫺3

0

共x ⫺ 1兲共2x 2 ⫹ 5x ⫺ 3兲 ⫽ 2x 3 ⫹ 3x 2 ⫺ 8x ⫹ 3

Finally, by factoring the quadratic, 2x2 ⫹ 5x ⫺ 3 ⫽ 共2x ⫺ 1兲共x ⫹ 3兲, you have 2x 3 ⫹ 3x 2 ⫺ 8x ⫹ 3 ⫽ 共x ⫺ 1兲共2x ⫺ 1兲共x ⫹ 3兲 and you can conclude that the zeros are x ⫽ 1, x ⫽ 12, and x ⫽ ⫺3.

✓CHECKPOINT 4 Find all real zeros of the polynomial. 2x3 ⫺ 3x2 ⫺ 3x ⫹ 2



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24

CHAPTER 0

A Precalculus Review

Exercises 0.4

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 8, use the Quadratic Formula to find all real zeros of the second-degree polynomial. 1.

6x 2

⫺ 7x ⫹ 1

2.

8x 2

⫺ 2x ⫺ 1

3. 4x 2 ⫺ 12x ⫹ 9

4. 9x 2 ⫹ 12x ⫹ 4

5. y 2 ⫹ 4y ⫹ 1

6. y2 ⫹ 5y ⫺ 2

7. 2x 2 ⫹ 3x ⫺ 4

8. 3x 2 ⫺ 8x ⫺ 4

In Exercises 9–18, write the second-degree polynomial as the product of two linear factors. 9. x2 ⫺ 4x ⫹ 4

10. x 2 ⫹ 10x ⫹ 25

11. 4x 2 ⫹ 4x ⫹ 1

12. 9x 2 ⫺ 12x ⫹ 4

13. 3x2 ⫺ 4x ⫹ 1

14. 2x 2 ⫺ x ⫺ 1

15. 3x 2 ⫺ 5x ⫹ 2

16. x 2 ⫺ xy ⫺ 2y 2

17. x 2 ⫺ 4xy ⫹ 4y 2

18. a 2 b 2 ⫺ 2abc ⫹ c 2

57. 冪x 2 ⫺ 7x ⫹ 12

58. 冪x 2 ⫺ 8x ⫹ 15

59.

60. 冪3x2 ⫺ 10x ⫹ 3

冪5x2

⫹ 6x ⫹ 1

In Exercises 61– 64, use synthetic division to complete the indicated factorization. 61. x 3 ⫺ 3x 2 ⫺ 6x ⫺ 2 ⫽ 共x ⫹ 1兲共 62.

x3



2x 2

⫺ x ⫹ 2 ⫽ 共x ⫺ 2兲共

63. 2x3 ⫺ x2 ⫺ 2x ⫹ 1 ⫽ 共x ⫹ 1)共 64.

x4



16x 3



96x 2

兲 兲 兲

⫺ 256x ⫹ 256 ⫽ 共x ⫺ 4兲共



In Exercises 65–74, use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. 65. x 3 ⫺ x 2 ⫺ 10x ⫺ 8

66. x 3 ⫺ 7x ⫺ 6

67. x 3 ⫺ 6x 2 ⫹ 11x ⫺ 6

68. x 3 ⫹ 2x 2 ⫺ 5x ⫺ 6

69. 6x 3 ⫺ 11x 2 ⫺ 19x ⫺ 6

70. 18x 3 ⫺ 9x 2 ⫺ 8x ⫹ 4

x3

72. 2x 3 ⫺ x 2 ⫺ 13x ⫺ 6

In Exercises 19–34, completely factor the polynomial.

71.

19. 81 ⫺ y 4

20. x 4 ⫺ 16

73. 4x3 ⫹ 11x2 ⫹ 5x ⫺ 2

21. x 3 ⫺ 8

22. y 3 ⫺ 64

23.

24. z 3 ⫹ 125

75. Production Level The minimum average cost of producing x units of a product occurs when the production level is set at the (positive) solution of

y3

⫹ 64



3x 2

⫺ 3x ⫺ 4

74. 3x3 ⫹ 4x2 ⫺ 13x ⫹ 6

25. x3 ⫺ y3

26. 共x ⫺ a兲 3 ⫹ b 3

27. x 3 ⫺ 4x 2 ⫺ x ⫹ 4

28. x 3 ⫺ x 2 ⫺ x ⫹ 1

0.0003x 2 ⫺ 1200 ⫽ 0.

29. 2x 3 ⫺ 3x 2 ⫹ 4x ⫺ 6

30. x 3 ⫺ 5x 2 ⫺ 5x ⫹ 25

31. 2x 3 ⫺ 4x 2 ⫺ x ⫹ 2

32. x 3 ⫺ 7x 2 ⫺ 4x ⫹ 28

How many solutions does this equation have? Find and interpret the solution(s) in the context of the problem. What production level will minimize the average cost?

33. x 4 ⫺ 15x 2 ⫺ 16

34. 2x 4 ⫺ 49x 2 ⫺ 25

In Exercises 35–54, find all real zeros of the polynomial. 35. x 2 ⫺ 5x

36. 2x 2 ⫺ 3x

37. x 2 ⫺ 9

38. x 2 ⫺ 25 40. x 2 ⫺ 8

39. x 2 ⫺ 3 41. 共x ⫺ 3兲 ⫺ 9

42. 共x ⫹ 1兲 2 ⫺ 36

43. x 2 ⫹ x ⫺ 2

44. x 2 ⫹ 5x ⫹ 6

45. x 2 ⫺ 5x ⫺ 6

46. x 2 ⫹ x ⫺ 20

47. 3x2 ⫹ 5x ⫹ 2

48. 2x2 ⫺ x ⫺ 1

2

49.

x3

51.

x4

⫺ 16

53.

x3

x2

⫹ 64 ⫺

⫺ 4x ⫹ 4

50.

x3

⫺ 216

52.

x4

⫺ 625

54.

2x 3

⫹ x 2 ⫹ 6x ⫹ 3

In Exercises 55–60, find the interval (or intervals) on which the given expression is defined. 55. 冪x 2 ⫺ 4 The symbol

56. 冪4 ⫺ x 2

76. Profit

The profit P from sales is given by

P ⫽ ⫺200x 2 ⫹ 2000x ⫺ 3800 where x is the number of units sold per day (in hundreds). Determine the interval for x such that the profit will be greater than 1000. 77. Chemistry: Finding Concentrations Quadratic Formula to solve the expression 1.8 ⫻ 10⫺5 ⫽

Use

the

x2 1.0 ⫻ 10⫺4 ⫺ x

which is needed to determine the quantity of hydrogen ions 共关H ⫹ 兴兲 in a solution of 1.0 ⫻ 10⫺4 M acetic acid. Because x represents a concentration of 关H ⫹ 兴, only positive values of x are possible solutions. (Source: Adapted from Zumdahl, Chemistry, Seventh Edition) 78. Finance After 2 years, an investment of $1200 is made at an interest rate r, compounded annually, that will yield an amount of A ⫽ 1200共1 ⫹ r兲 2. Determine the interest rate if A ⫽ $1300.

indicates an exercise that contains material from textbooks in other disciplines.

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SECTION 0.5

Fractions and Rationalization

25

Section 0.5

Fractions and Rationalization

■ Add and subtract rational expressions. ■ Simplify rational expressions involving radicals. ■ Rationalize numerators and denominators of rational expressions.

Operations with Fractions In this section, you will review operations involving fractional expressions such as 2 , x

x 2 ⫹ 2x ⫺ 4 , x⫹6

1

and

冪x 2 ⫹ 1

.

The first two expressions have polynomials as both numerator and denominator and are called rational expressions. A rational expression is proper if the degree of the numerator is less than the degree of the denominator. For example, x x2 ⫹ 1 is proper. If the degree of the numerator is greater than or equal to the degree of the denominator, then the rational expression is improper. For example, x2

x2 , ⫹1

and

x 3 ⫹ 2x ⫹ 1 x⫹1

are both improper. Operations with Fractions

1. Add fractions (find a common denominator): c b ad a c a d bc ad ⫹ bc ⫹ ⫽ ⫹ ⫽ ⫹ ⫽ , b d b d d b bd bd bd 2. Subtract fractions (find a common denominator):

冢冣

冢冣

c b ad a c a d bc ad ⫺ bc ⫺ ⫽ ⫺ ⫽ ⫺ ⫽ , b d b d d b bd bd bd 3. Multiply fractions:

冢冣

冢ab冣冢dc 冣 ⫽ bdac ,

冢冣

b ⫽ 0, d ⫽ 0

b ⫽ 0, d ⫽ 0

b ⫽ 0, d ⫽ 0

4. Divide fractions (invert and multiply): a a兾b ⫽ c兾d b

冢 冣冢dc冣 ⫽ adbc,

a a兾b a兾b ⫽ ⫽ c c兾1 b

冢 冣冢1c 冣 ⫽ bca ,

b ⫽ 0,

c ⫽ 0, d ⫽ 0 5. Divide out like factors: ab b ⫽ , ac c

ab ⫹ ac a共b ⫹ c兲 b ⫹ c ⫽ ⫽ , ad ad d

a ⫽ 0, c ⫽ 0, d ⫽ 0

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

26

CHAPTER 0

A Precalculus Review

Example 1

Adding and Subtracting Rational Expressions

Perform each indicated operation and simplify. a. x ⫹

1 x

b.

1 2 ⫺ x ⫹ 1 2x ⫺ 1

SOLUTION

a. x ⫹

b.

1 x2 1 ⫽ ⫹ x x x 2 ⫹ 1 x ⫽ x

Write with common denominator. Add fractions.

共2x ⫺ 1兲 1 2(x ⫹ 1兲 2 ⫺ ⫺ ⫽ x ⫹ 1 2x ⫺ 1 共x ⫹ 1兲共2x ⫺ 1兲 共x ⫹ 1兲共2x ⫺ 1兲 ⫽

2x ⫺ 1 ⫺ 2x ⫺ 2 ⫺3 ⫽ 2 2x 2 ⫹ x ⫺ 1 2x ⫹ x ⫺ 1

✓CHECKPOINT 1 Perform each indicated operation and simplify. a. x ⫹

2 x

b.

2 1 ⫺ x ⫹ 1 2x ⫹ 1



In adding (or subtracting) fractions whose denominators have no common factors, it is convenient to use the following pattern. a c a ⫹ ⫽ b d b



c

d



ad ⫹ bc bd

For instance, in Example 1(b), you could have used this pattern as shown.

共2x ⫺ 1兲 ⫺ 2共x ⫹ 1兲 1 2 ⫺ ⫽ x ⫹ 1 2x ⫺ 1 共x ⫹ 1兲共2x ⫺ 1兲 ⫽

2x ⫺ 1 ⫺ 2x ⫺ 2 ⫺3 ⫽ 2 共x ⫹ 1兲共2x ⫺ 1兲 2x ⫹ x ⫺ 1

In Example 1, the denominators of the rational expressions have no common factors. When the denominators do have common factors, it is best to find the least common denominator before adding or subtracting. For instance, when adding 1兾x and 2兾x 2, you can recognize that the least common denominator is x 2 and write 1 x 2 2 ⫹ 2⫽ 2⫹ 2 x x x x ⫽

x ⫹ 2. x2

Write with common denominator. Add fractions.

This is further demonstrated in Example 2.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 0.5

Example 2

27

Fractions and Rationalization

Adding and Subtracting Rational Expressions

Perform each indicated operation and simplify. a.

x 3 ⫹ x2 ⫺ 1 x ⫹ 1

b.

1 1 ⫺ 2共x 2 ⫹ 2x兲 4x

SOLUTION

a. Because x 2 ⫺ 1 ⫽ 共x ⫹ 1兲共x ⫺ 1兲, the least common denominator is x 2 ⫺ 1. x2

x 3 3 x ⫹ ⫹ ⫽ ⫺ 1 x ⫹ 1 共x ⫺ 1兲共x ⫹ 1兲 x ⫹ 1 x 3共x ⫺ 1兲 ⫽ ⫹ 共x ⫺ 1兲共x ⫹ 1兲 共x ⫺ 1兲共x ⫹ 1兲

Factor. Write with common denominator.



x ⫹ 3x ⫺ 3 共x ⫺ 1兲共x ⫹ 1兲

Add fractions.



4x ⫺ 3 x2 ⫺ 1

Simplify.

b. In this case, the least common denominator is 4x共x ⫹ 2兲. 1 1 1 1 ⫺ ⫺ ⫽ 2共x 2 ⫹ 2x兲 4x 2x共x ⫹ 2兲 2共2x兲

Factor.



2 x⫹2 ⫺ 2共2x兲共x ⫹ 2兲 2共2x兲共x ⫹ 2兲

Write with common denominator.



2⫺x⫺2 4x共x ⫹ 2兲

Subtract fractions.



⫺x 4x共x ⫹ 2兲

Divide out like factor.



⫺1 , 4共x ⫹ 2兲

x⫽0

Simplify.

✓CHECKPOINT 2 Perform each indicated operation and simplify. a.

x 2 ⫹ x2 ⫺ 4 x ⫺ 2

b.

1 1 ⫺ 3共x2 ⫹ 2x兲 3x



STUDY TIP To add more than two fractions, you must find a denominator that is common to all the fractions. For instance, to add 12, 13, and 15, use a (least) common denominator of 30 and write 1 1 1 15 10 6 ⫹ ⫹ ⫽ ⫹ ⫹ 2 3 5 30 30 30 ⫽

31 . 30

Write with common denominator. Add fractions.

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28

CHAPTER 0

A Precalculus Review

To add more than two rational expressions, use a similar procedure, as shown in Example 3. (Expressions such as those shown in this example are used in calculus to perform an integration technique called integration by partial fractions.)

Example 3

Adding More than Two Rational Expressions

Perform each indicated addition of rational expressions. a.

A B C ⫹ ⫹ x⫹2 x⫺3 x⫹4

b.

A C B ⫹ ⫹ x ⫹ 2 共x ⫹ 2兲2 x ⫺ 1

SOLUTION

a. The least common denominator is 共x ⫹ 2兲共x ⫺ 3兲共x ⫹ 4兲. A B C ⫹ ⫹ x⫹2 x⫺3 x⫹4 A共x ⫺ 3兲共x ⫹ 4兲 ⫹ B共x ⫹ 2兲共x ⫹ 4兲 ⫹ C共x ⫹ 2兲共x ⫺ 3兲 ⫽ 共x ⫹ 2兲共x ⫺ 3兲共x ⫹ 4兲 2 A共x ⫹ x ⫺ 12兲 ⫹ B共x 2 ⫹ 6x ⫹ 8兲 ⫹ C共x 2 ⫺ x ⫺ 6兲 ⫽ 共x ⫹ 2兲共x ⫺ 3兲共x ⫹ 4兲 2 ⫹ Bx 2 ⫹ Cx 2 ⫹ Ax ⫹ 6Bx ⫺ Cx ⫺ 12A ⫹ 8B ⫺ 6C Ax ⫽ 共x ⫹ 2兲共x ⫺ 3兲共x ⫹ 4兲 2 共A ⫹ B ⫹ C兲x ⫹ 共A ⫹ 6B ⫺ C兲 x ⫹ 共⫺12A ⫹ 8B ⫺ 6C兲 ⫽ 共x ⫹ 2兲共x ⫺ 3兲共x ⫹ 4兲 b. Here the least common denominator is 共x ⫹ 2兲 2 共x ⫺ 1兲. A C B ⫹ ⫹ x ⫹ 2 共x ⫹ 2兲2 x ⫺ 1 A共x ⫹ 2兲共x ⫺ 1兲 ⫹ B共x ⫺ 1兲 ⫹ C共x ⫹ 2兲 2 ⫽ 共x ⫹ 2兲 2共x ⫺ 1兲 A共x 2 ⫹ x ⫺ 2兲 ⫹ B共x ⫺ 1兲 ⫹ C共x 2 ⫹ 4x ⫹ 4兲 ⫽ 共x ⫹ 2兲 2共x ⫺ 1兲 Ax 2 ⫹ Cx 2 ⫹ Ax ⫹ Bx ⫹ 4Cx ⫺ 2A ⫺ B ⫹ 4C ⫽ 共x ⫹ 2兲 2共x ⫺ 1兲 共A ⫹ C兲x 2 ⫹ 共A ⫹ B ⫹ 4C兲x ⫹ 共⫺2A ⫺ B ⫹ 4C兲 ⫽ 共x ⫹ 2兲 2共x ⫺ 1兲

✓CHECKPOINT 3 Perform each indicated addition of rational expressions. a.

A B C ⫹ ⫹ x⫹1 x⫺1 x⫹2

b.

A C B ⫹ ⫹ x ⫹ 1 共x ⫹ 1兲2 x ⫺ 2



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 0.5

29

Fractions and Rationalization

Expressions Involving Radicals In calculus, the operation of differentiation tends to produce “messy” expressions when applied to fractional expressions. This is especially true when the fractional expressions involve radicals. When differentiation is used, it is important to be able to simplify these expressions so that you can obtain more manageable forms. All of the expressions in Examples 4 and 5 are the results of differentiation. In each case, note how much simpler the simplified form is than the original form.

Example 4

Simplifying an Expression with Radicals

Simplify each expression. x 2冪x ⫹ 1 x⫹1

冪x ⫹ 1 ⫺

a.

b.

冢 x ⫹ 冪1x

⫹1

2

冣冢1 ⫹ 2冪x2x⫹ 1冣 2

SOLUTION

x 2共x ⫹ 1兲 x ⫺ 2冪x ⫹ 1 2冪x ⫹ 1 2冪x ⫹ 1 ⫽ x⫹1 x⫹1 2x ⫹ 2 ⫺ x 2冪x ⫹ 1 ⫽ x⫹1 1 1 x⫹2 ⫽ 2冪x ⫹ 1 x ⫹ 1 x⫹2 ⫽ 2共x ⫹ 1兲3兾2

冪x ⫹ 1 ⫺

a.



b.

冢 x ⫹ 冪1x

2



Write with common denominator.

Subtract fractions.

To divide, invert and multiply Multiply.

2x 1⫹ 冣冢 冪 ⫹1 2 x ⫹ 1冣 1 x ⫽冢 1⫹ 冣冢 冪 冪 x⫹ x ⫹1 x ⫹ 1冣 2

2



冢 x ⫹ 冪1x



冢x ⫹ 冪1x



2

x ⫹ 1 冣 冢 冪x 冪

2

2 2

⫹1 x ⫹ ⫹ 1 冪x 2 ⫹ 1

x⫹ x ⫹1 冣 冢 冪x ⫹ 1 冣 ⫹1 冪

2



2

2

1 ⫹1

冪x 2

✓CHECKPOINT 4 Simplify each expression. x 4冪x ⫹ 2 x⫹2

冪x ⫹ 2 ⫺

a.

b.

冢x ⫹

1 冪x2 ⫹ 4

冣冢1 ⫹

x 冪x2 ⫹ 4





Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

30

CHAPTER 0

A Precalculus Review

Example 5

Simplifying an Expression with Radicals

Simplify the expression. ⫺x

冢2冪x2x⫹ 1冣 ⫹ 冪x

2

2

⫹1 ⫹

x2

冢 x ⫹ 冪1x

2x 1⫹ 冣冢 冪 ⫹1 2 x ⫹ 1冣

2

2

From Example 4(b), you already know that the second part of this sum simplifies to 1兾冪x 2 ⫹ 1. The first part simplifies as shown. SOLUTION

⫺x

冢2冪x2x⫹ 1冣 ⫹ 冪x

2

2

⫹1

x2

⫽ ⫽

冪x 2 ⫹ 1 ⫺x 2 ⫹ x2 x 2冪x 2 ⫹ 1

⫺x 2 x2 ⫹ 1 ⫹ 2 2 ⫹ 1 x 冪x ⫹ 1

x 2冪x 2

⫺x 2 ⫹ x 2 ⫹ 1 x 2冪x 2 ⫹ 1 1 ⫽ 2 2 x 冪x ⫹ 1 ⫽

So, the sum is ⫺x

冢2冪x2x⫹ 1冣 ⫹ 冪x

2

2

⫹1

x2



冢 x ⫹ 冪1x



1 1 ⫹ x 2冪x 2 ⫹ 1 冪x 2 ⫹ 1



1 x2 ⫹ x 2冪x 2 ⫹ 1 x 2冪x 2 ⫹ 1



x2 ⫹ 1 x 2冪x 2 ⫹ 1



⫹1

2

冣冢1 ⫹ 2冪x2x⫹ 1冣 2

冪x 2 ⫹ 1 .

x2

✓CHECKPOINT 5 Simplify the expression. ⫺x

冢 3冪x3x⫹ 4 冣 ⫹ 冪x 2

x2

2

⫹4 ⫹

冢x ⫹ 冪1x

2

⫹4

冣冢1 ⫹ 3冪x3x⫹ 4冣 ■ 2

STUDY TIP To check that the simplified expression in Example 5 is equivalent to the original expression, try substituting values of x into each expression. For instance, when you substitute x ⫽ 1 into each expression, you obtain 冪2.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 0.5

Fractions and Rationalization

31

Rationalization Techniques In working with quotients involving radicals, it is often convenient to move the radical expression from the denominator to the numerator, or vice versa. For example, you can move 冪2 from the denominator to the numerator in the following quotient by multiplying by 冪2兾冪2. Radical in Denominator

Rationalize

Radical in Numerator

冢 冣

1 冪2

1 冪2 冪2 冪2

冪2

2

This process is called rationalizing the denominator. A similar process is used to rationalize the numerator. STUDY TIP The success of the second and third rationalizing techniques stems from the following.

共冪a ⫺ 冪b 兲共冪a ⫹ 冪b 兲

Rationalizing Techniques

1. If the denominator is 冪a, multiply by

冪a . 冪a

2. If the denominator is 冪a ⫺ 冪b, multiply by

冪a ⫹ 冪b . 冪a ⫹ 冪b

3. If the denominator is 冪a ⫹ 冪b, multiply by

冪a ⫺ 冪b . 冪a ⫺ 冪b

⫽a⫺b

The same guidelines apply to rationalizing numerators.

Example 6

Rationalizing Denominators and Numerators

Rationalize the denominator or numerator. a.

3 冪12

b.

冪x ⫹ 1

c.

2

1 冪5 ⫹ 冪2

d.

1 冪x ⫺ 冪x ⫹ 1

SOLUTION

a.

✓CHECKPOINT 6 Rationalize the denominator or numerator. 5 a. 冪8 b.

冪x ⫹ 2

4 1 c. 冪6 ⫺ 冪3 1 d. 冪x ⫹ 冪x ⫹ 2

b. c. d.

冢 冣 ⫽ 32共33兲 ⫽ 23 x⫹1 x⫹1 x⫹1 x⫹1 ⫽ ⫽ 冢 冣 2 2 2 x⫹1 x⫹1 1 5⫺ 2 5⫺ 2 1 ⫽ ⫽ ⫽ 冢 冣 ⫺ 5⫺2 5⫹ 2 5⫹ 2 5 2 x⫹ x⫹1 1 1 ⫽ 冢 x⫺ x⫹1 x⫺ x⫹1 x ⫹ x ⫹ 1冣

3 3 3 冪3 ⫽ ⫽ 冪12 2冪3 2冪3 冪3 冪

























⫽ ■

























冪5 ⫺ 冪2

3

冪x ⫹ 冪x ⫹ 1

x ⫺ 共x ⫹ 1兲

⫽ ⫺ 冪x ⫺ 冪x ⫹ 1

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

32

CHAPTER 0

A Precalculus Review

Exercises 0.5

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–16, perform the indicated operations and simplify your answer.

28.

⫺x 3 ⫹ 2共3 ⫹ x 2兲3兾2 共3 ⫹ x 2兲1兾2

1.

x 3 ⫹ x⫺2 x⫺2

2.

2x ⫺ 1 1 ⫺ x ⫹ x⫹3 x⫹3

3.

2x 1 ⫺ 3x ⫺ 2 x2 ⫹ 2 x ⫹2

4.

5x ⫹ 10 2x ⫹ 10 ⫺ 2x ⫺ 1 2x ⫺ 1

5.

2 1 ⫺ x2 ⫺ 4 x ⫺ 2

6.

x 1 ⫺ x2 ⫹ x ⫺ 2 x ⫹ 2

31.

7.

5 3 ⫹ x⫺3 3⫺x

8.

x 2 ⫹ 2⫺x x⫺2

33.

9.

A C B ⫹ ⫹ x ⫺ 1 共x ⫺ 1兲 2 x ⫹ 2

35.

10.

A B C ⫹ ⫹ x ⫺ 5 x ⫹ 5 共x ⫹ 5兲 2

37.

38.

11.

A Bx ⫹ C ⫹ 2 x⫺6 x ⫹3

2x 5 ⫺ 冪3

39.

1 冪6 ⫹ 冪5

40.

12.

29.

Ax ⫹ B C ⫹ x2 ⫹ 2 x⫺4

2 1⫺x ⫹ 14. x ⫹ 1 x 2 ⫺ 2x ⫹ 3

2 1 13. ⫺ ⫹ 2 x x ⫹2 15.

In Exercises 29– 44, rationalize the numerator or denominator and simplify.

41.

1 x ⫺ x 2 ⫺ x ⫺ 2 x 2 ⫺ 5x ⫹ 6

43.

x⫺1 2 10 ⫹ ⫹ 16. 2 x ⫹ 5x ⫹ 4 x 2 ⫺ x ⫺ 2 x 2 ⫹ 2x ⫺ 8

19.

2⫺t 2冪1 ⫹ t

⫺ 冪1 ⫹ t

冢 22. 冢

21. 2x冪x 2 ⫹ 1 ⫺ 冪x3 ⫹ 1 ⫺

23. 24.

x2



⫹ 2兲

1兾2



x3 ⫼ 共x 2 ⫹ 1兲 冪x 2 ⫹ 1 3x 3

2冪x 3 ⫹ 1 x2

x2

⫺ 共 x2

冣 ⫼ 共x

⫹ 2兲

⫺1兾2

x共x ⫹ 1兲⫺1兾2 ⫺ 共x ⫹ 1兲1兾2 x2 冪x ⫹ 1

25.

18. 2冪x 共x ⫺ 2兲 ⫹

冪x



冪x 冪x ⫹ 1

2共x ⫹ 1兲

3

⫹ 1兲

4x

32.

冪x ⫺ 1

49共x ⫺ 3兲 冪x 2 ⫺ 9

34.

5

36.

冪14 ⫺ 2

2

42.

冪x ⫹ 冪x ⫺ 2 冪x ⫹ 2 ⫺ 冪2

44.

x

冪4 ⫺ x 2

共x ⫺ 2兲 2冪x 冪x 2 ⫹ 1 1 ⫹ 20. ⫺ 2 2 冪 x x ⫹1

⫺x 2 ⫹ 共x ⫹ 1兲 3兾2 共x ⫹ 1兲1兾2

30.

3 冪21

5y 冪y ⫹ 7

10共x ⫹ 2兲 冪x 2 ⫺ x ⫺ 6

13 6 ⫹ 冪10 x 冪2 ⫹ 冪3 冪15 ⫹ 3

12 10 冪x ⫹ 冪x ⫹ 5 冪x ⫹ 1 ⫺ 1

x

In Exercises 45 and 46, perform the indicated operations and rationalize as needed.

In Exercises 17–28, simplify each expression. 17.

2 冪10

2

45.

冪x 2 ⫹ 1

2 x 2冪4 ⫺ x 2 4 ⫺ x2

x4



46.

x2

1 x冪x 2 ⫹ 1 x2 ⫹ 1 ⫺

47. Installment Loan The monthly payment M for an installment loan is given by the formula M⫽P



r兾12 1 1⫺ 共r兾12兲 ⫹ 1





N



where P is the amount of the loan, r is the annual percentage rate, and N is the number of monthly payments. Enter the formula into a graphing utility, and use it to find the monthly payment for a loan of $10,000 at an annual percentage rate of 7.5% 共r ⫽ 0.075兲 for 5 years 共N ⫽ 60 monthly payments兲. 48. MAKE A DECISION: INVENTORY A retailer has determined that the cost C of ordering and storing x units of a product is 900,000 . x

2x 2 ⫺ 共x 2 ⫺ 1兲1兾3 3共 ⫺ 1兲 2兾3 26. x2

(a) Write the expression for cost as a single fraction.

⫺x 2 2x ⫹ 27. 共2x ⫹ 3兲 3兾2 共 2x ⫹ 3兲1兾2

(b) Which order size should the retailer place: 240 units, 387 units, or 480 units? Explain your reasoning.

x2

C ⫽ 6x ⫹

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

1

AP/Wide World Photos

Functions, Graphs, and Limits

A graph showing changes in a company’s earnings and other financial indicators can depict the company’s general financial trends over time. (See Section 1.2, Example 8.)

Applications Functions and limit concepts have many real-life applications. The applications listed below represent a sample of the applications in this chapter. ■ ■ ■ ■ ■ ■

1.1

1.2 1.3 1.4 1.5 1.6

The Cartesian Plane and the Distance Formula Graphs of Equations Lines in the Plane and Slope Functions Limits Continuity

Health, Exercise 36, page 42 Federal Education Spending, Exercise 70, page 55 Profit Analysis, Exercise 93, page 67 Make a Decision: Choosing a Job, Exercise 95, page 67 Prescription Drugs, Exercise 63, page 80 Consumer Awareness, Exercise 61, page 104 33 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

34

CHAPTER 1

Functions, Graphs, and Limits

Section 1.1

The Cartesian Plane and the Distance Formula

■ Plot points in a coordinate plane and read data presented graphically. ■ Find the distance between two points in a coordinate plane. ■ Find the midpoints of line segments connecting two points. ■ Translate points in a coordinate plane.

The Cartesian Plane y-axis

Vertical real line

4 3

Quadrant II

Quadrant I

2

Origin 1

Horizontal real line

−4 − 3 −2 − 1 −1

1

−2

Quadrant III

2

3

x-axis

4

Quadrant IV

−3 −4

FIGURE 1.1

The Cartesian Plane

Just as you can represent real numbers by points on a real number line, you can represent ordered pairs of real numbers by points in a plane called the rectangular coordinate system, or the Cartesian plane, after the French mathematician René Descartes (1596–1650). The Cartesian plane is formed by using two real number lines intersecting at right angles, as shown in Figure 1.1. The horizontal real number line is usually called the x-axis, and the vertical real number line is usually called the y-axis. The point of intersection of these two axes is the origin, and the two axes divide the plane into four parts called quadrants. Each point in the plane corresponds to an ordered pair 共x, y兲 of real numbers x and y, called coordinates of the point. The x-coordinate represents the directed distance from the y-axis to the point, and the y-coordinate represents the directed distance from the x-axis to the point, as shown in Figure 1.2.

共x, y兲

y-axis

x

Directed distance from y-axis

(x, y) y x-axis

Directed distance from x-axis

STUDY TIP The notation 共x, y兲 denotes both a point in the plane and an open interval on the real number line. The context will tell you which meaning is intended.

FIGURE 1.2

Example 1

Plot the points 共⫺1, 2兲, 共3, 4兲, 共0, 0兲, 共3, 0兲, and 共⫺2, ⫺3兲.

y

(3, 4)

4

SOLUTION

To plot the point

3

共⫺1, 2兲

(−1, 2) 1 −4 − 3 − 2 − 1 −1 −2

(−2, −3)

−3 −4

FIGURE 1.3

Plotting Points in the Cartesian Plane

x-coordinate (0, 0) 1

(3, 0) 2

3

x 4

y-coordinate

imagine a vertical line through ⫺1 on the x-axis and a horizontal line through 2 on the y-axis. The intersection of these two lines is the point 共⫺1, 2兲. The other four points can be plotted in a similar way and are shown in Figure 1.3.

✓CHECKPOINT 1 Plot the points 共⫺3, 2兲, 共4, ⫺2兲, 共3, 1兲, 共0, ⫺2兲, and 共⫺1, ⫺2兲.



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.1

35

The Cartesian Plane and the Distance Formula

Using a rectangular coordinate system allows you to visualize relationships between two variables. In Example 2, notice how much your intuition is enhanced by the use of a graphical presentation.

Example 2

Amounts Spent on Snowmobiles

Dollars (in millions)

A

Sketching a Scatter Plot

1200

The amounts A (in millions of dollars) spent on snowmobiles in the United States from 1997 through 2006 are shown in the table, where t represents the year. Sketch a scatter plot of the data. (Source: International Snowmobile Manufacturers

1000

Association)

800 600

1997 1999 2001 2003 2005

t

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

A

1006

975

883

821

894

817

779

712

826

741

t

Year

FIGURE 1.4

STUDY TIP In Example 2, you could let t ⫽ 1 represent the year 1997. In that case, the horizontal axis would not have been broken, and the tick marks would have been labeled 1 through 10 (instead of 1997 through 2006).

To sketch a scatter plot of the data given in the table, you simply represent each pair of values by an ordered pair 共t, A兲, and plot the resulting points, as shown in Figure 1.4. For instance, the first pair of values is represented by the ordered pair (1997, 1006). Note that the break in the t-axis indicates that the numbers between 0 and 1996 have been omitted.

SOLUTION

✓CHECKPOINT 2 From 1995 through 2004, the enrollments E (in millions) of students in U.S. public colleges are shown, where t represents the year. Sketch a scatter plot of the data. (Source: U.S. National Center for Education Statistics) t

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

E

11.1

11.1

11.2

11.1

11.3

11.8

12.2

12.8

12.9

13.0 ■

TECHNOLOGY Amounts Spent on Snowmobiles

Amounts Spent on Snowmobiles

1200

1200

A

1000 800 600

1997 1999 2001 2003 2005

Year

The symbol

A

Dollars (in millions)

Dollars (in millions)

The scatter plot in Example 2 is only one way to represent the given data graphically. Two other techniques are shown at the right. The first is a bar graph and the second is a line graph. All three graphical representations were created with a computer. If you have access to computer graphing software, try using it to represent graphically the data given in Example 2.

t

1000 800 600

1997 1999 2001 2003 2005

Year

indicates an example that uses or is derived from real-life data.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

t

36

CHAPTER 1

Functions, Graphs, and Limits

The Distance Formula

a2 + b2 = c2

Recall from the Pythagorean Theorem that, for a right triangle with hypotenuse of length c and sides of lengths a and b, you have

c

a

a2 ⫹ b2 ⫽ c2

b

FIGURE 1.5

Pythagorean Theorem

Pythagorean Theorem

as shown in Figure 1.5. (The converse is also true. That is, if a 2 ⫹ b 2 ⫽ c 2, then the triangle is a right triangle.) Suppose you want to determine the distance d between two points 共x1, y1兲 and 共x2, y2兲 in the plane. With these two points, a right triangle can be formed, as shown in Figure 1.6. The length of the vertical side of the triangle is

ⱍy2 ⫺ y1ⱍ and the length of the horizontal side is

y

ⱍx2 ⫺ x1ⱍ.

(x1, y1)

y1

By the Pythagorean Theorem, you can write d

⏐y2 − y1⏐



(x2, y2)

y2

x

x2

x1













This result is the Distance Formula.

⏐x2 − x1⏐

FIGURE 1.6 Two Points



d 2 ⫽ x2 ⫺ x1 2 ⫹ y2 ⫺ y1 2 d ⫽ 冪 x2 ⫺ x1 2 ⫹ y2 ⫺ y1 2 d ⫽ 冪共x2 ⫺ x1兲2 ⫹ 共 y2 ⫺ y1兲2.

Distance Between

The Distance Formula

The distance d between the points 共x1, y1兲 and 共x2, y2兲 in the plane is d ⫽ 冪共x2 ⫺ x1兲 2 ⫹ 共 y2 ⫺ y1兲2.

Example 3

Find the distance between the points 共⫺2, 1兲 and 共3, 4兲.

y 4 3

Let 共x1, y1兲 ⫽ 共⫺2, 1兲 and 共x2, y2兲 ⫽ 共3, 4兲. Then apply the Distance Formula as shown. SOLUTION

(3, 4) d 3

(−2, 1) 5 −3

−2

−1

1 −1

FIGURE 1.7

Finding a Distance

2

3

4

x

d ⫽ 冪共x2 ⫺ x1兲 2 ⫹ 共 y2 ⫺ y1兲2 ⫽ 冪关3 ⫺ 共⫺2兲兴2 ⫹ 共4 ⫺ 1兲2 ⫽ 冪共5兲2 ⫹ 共3兲2 ⫽ 冪34 ⬇ 5.83

Distance Formula Substitute for x1, y1, x2, and y2. Simplify.

Use a calculator.

Note in Figure 1.7 that a distance of 5.83 looks about right.

✓CHECKPOINT 3 Find the distance between the points 共⫺2, 1兲 and 共2, 4兲.



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.1

Example 4

y

8

The Cartesian Plane and the Distance Formula

37

Verifying a Right Triangle

Use the Distance Formula to show that the points 共2, 1兲, 共4, 0兲, and 共5, 7兲 are vertices of a right triangle.

(5, 7)

The three points are plotted in Figure 1.8. Using the Distance Formula, you can find the lengths of the three sides as shown below. SOLUTION

6

d1 ⫽ 冪共5 ⫺ 2兲2 ⫹ 共7 ⫺ 1兲2 ⫽ 冪9 ⫹ 36 ⫽ 冪45 d2 ⫽ 冪共4 ⫺ 2兲2 ⫹ 共0 ⫺ 1兲2 ⫽ 冪4 ⫹ 1 ⫽ 冪5 d3 ⫽ 冪共5 ⫺ 4兲2 ⫹ 共7 ⫺ 0兲2 ⫽ 冪1 ⫹ 49 ⫽ 冪50

d1

4

d3 2

Because

(2, 1)

d2 2

(4, 0) 4

x

6

d12 ⫹ d 22 ⫽ 45 ⫹ 5 ⫽ 50 ⫽ d 32 you can apply the converse of the Pythagorean Theorem to conclude that the triangle must be a right triangle.

FIGURE 1.8

✓CHECKPOINT 4 Use the Distance Formula to show that the points 共2, ⫺1兲, 共5, 5兲, and 共6, ⫺3兲 are vertices of a right triangle. ■ The figures provided with Examples 3 and 4 were not really essential to the solution. Nevertheless, we strongly recommend that you develop the habit of including sketches with your solutions—even if they are not required.

(50, 45)

Example 5

Finding the Length of a Pass

In a football game, a quarterback throws a pass from the 5-yard line, 20 yards from the sideline. The pass is caught by a wide receiver on the 45-yard line, 50 yards from the same sideline, as shown in Figure 1.9. How long was the pass? You can find the length of the pass by finding the distance between the points 共20, 5兲 and 共50, 45兲. SOLUTION Line of scrimmage (20, 5) 10

20

30

40

50

FIGURE 1.9

✓CHECKPOINT 5 A quarterback throws a pass from the 10-yard line, 10 yards from the sideline. The pass is caught by a wide receiver on the 30-yard line, 25 yards from the same sideline. How long was the pass? ■

d ⫽ 冪共50 ⫺ 20兲2 ⫹ 共45 ⫺ 5兲2 ⫽ 冪900 ⫹ 1600 ⫽ 50

Distance Formula

Simplify.

So, the pass was 50 yards long.

STUDY TIP In Example 5, the scale along the goal line showing distance from the sideline does not normally appear on a football field. However, when you use coordinate geometry to solve real-life problems, you are free to place the coordinate system in any way that is convenient to the solution of the problem.

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38

CHAPTER 1

Functions, Graphs, and Limits

The Midpoint Formula To find the midpoint of the line segment that joins two points in a coordinate plane, you can simply find the average values of the respective coordinates of the two endpoints. The Midpoint Formula

The midpoint of the segment joining the points 共x1, y1兲 and 共x2, y2兲 is Midpoint ⫽

y



x1 ⫹ x2 y1 ⫹ y2 . , 2 2



6

(2, 0) −6

−3

(−5, − 3)

Example 6

(9, 3)

3

3

6

9

Midpoint

−3

x

Find the midpoint of the line segment joining the points 共⫺5, ⫺3兲 and 共9, 3兲, as shown in Figure 1.10. SOLUTION

−6

Finding a Segment’s Midpoint

Let 共x1, y1兲 ⫽ 共⫺5, ⫺3兲 and 共x2, y2兲 ⫽ 共9, 3兲.

Midpoint ⫽

FIGURE 1.10



x1 ⫹ x 2 y1 ⫹ y2 ⫺5 ⫹ 9 ⫺3 ⫹ 3 ⫽ ⫽ 共2, 0兲 , , 2 2 2 2

冣 冢



✓CHECKPOINT 6 Find the midpoint of the line segment joining 共⫺6, 2兲 and 共2, 8兲. Starbucks Corporation’s Annual Sales

Sales (in billions of dollars)

7 6

(2005, 6.37) Midpoint

5 4

(2004, 5.23) (2003, 4.08)

3

2003

2004

Year

FIGURE 1.11

2005

Example 7



Estimating Annual Sales

Starbucks Corporation had annual sales of $4.08 billion in 2003 and $6.37 billion in 2005. Without knowing any additional information, what would you estimate the 2004 sales to have been? (Source: Starbucks Corp.) One solution to the problem is to assume that sales followed a linear pattern. With this assumption, you can estimate the 2004 sales by finding the midpoint of the segment connecting the points (2003, 4.08) and (2005, 6.37). SOLUTION

Midpoint ⫽

冢2003 ⫹2 2005, 4.08 ⫹2 6.37冣 ⬇ 共2004, 5.23兲

So, you would estimate the 2004 sales to have been about $5.23 billion, as shown in Figure 1.11. (The actual 2004 sales were $5.29 billion.)

✓CHECKPOINT 7 Whirlpool Corporation had annual sales of $12.18 billion in 2003 and $14.32 billion in 2005. What would you estimate the 2004 annual sales to have been? (Source: Whirlpool Corp.) ■

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.1

39

The Cartesian Plane and the Distance Formula

Translating Points in the Plane Example 8

Translating Points in the Plane

Figure 1.12(a) shows the vertices of a parallelogram. Find the vertices of the parallelogram after it has been translated two units down and four units to the right. To translate each vertex two units down, subtract 2 from each y-coordinate. To translate each vertex four units to the right, add 4 to each x-coordinate. SOLUTION

Original Point

Translated Point

共1, 0兲 共3, 2兲 共3, 6兲 共1, 4兲

共1 ⫹ 4, 0 ⫺ 2兲 ⫽ 共5, ⫺2兲 共3 ⫹ 4, 2 ⫺ 2兲 ⫽ 共7, 0兲 共3 ⫹ 4, 6 ⫺ 2兲 ⫽ 共7, 4兲 共1 ⫹ 4, 4 ⫺ 2兲 ⫽ 共5, 2兲

The translated parallelogram is shown in Figure 1.12(b). Walt Disney/The Kobal Collection

Many movies now use extensive computer graphics, much of which consists of transformations of points in two- and three-dimensional space. The photo above shows a character from Pirates of the Caribbean: Dead Man’s Chest. The movie’s animators used computer graphics to design the scenery, characters, motion, and even the lighting throughout much of the film.

8

8

(3, 6)

(3, 6) (3, 2)

(3, 2) (7, 4)

(1, 4) −6

(1, 4) (1, 0)

12

−6

(5, 2) (1, 0)

(7, 0)

12

(5, − 2) −4

(a)

−4

(b)

FIGURE 1.12

✓CHECKPOINT 8 Find the vertices of the parallelogram in Example 8 after it has been translated two units to the left and four units down. ■

CONCEPT CHECK 1. What is the y-coordinate of any point on the x-axis? What is the x-coordinate of any point on the y-axis? 2. Describe the signs of the x- and y-coordinates of points that lie in the first and second quadrants. 3. To divide a line segment into four equal parts, how many times is the Midpoint Formula used? 4. When finding the distance between two points, does it matter which point is chosen as 冇x1, y1冈? Explain.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

40

CHAPTER 1

Functions, Graphs, and Limits

Skills Review 1.1

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Section 0.3.

In Exercises 1–6, simplify each expression. 1. 冪共3 ⫺ 6兲2 ⫹ 关1 ⫺ 共⫺5兲兴2 3.

2. 冪共⫺2 ⫺ 0兲 2 ⫹ 关⫺7 ⫺ 共⫺3兲兴 2

5 ⫹ 共⫺4兲 2

4.

5. 冪27 ⫹ 冪12

⫺3 ⫹ 共⫺1兲 2

6. 冪8 ⫺ 冪18

In Exercises 7–10, solve for x or y. 7. 冪共3 ⫺ x兲2 ⫹ 共7 ⫺ 4兲 2 ⫽ 冪45 9.

8. 冪共6 ⫺ 2兲2 ⫹ 共⫺2 ⫺ y兲2 ⫽ 冪52

x ⫹ 共⫺5兲 ⫽7 2

10.

Exercises 1.1

⫺7 ⫹ y ⫽ ⫺3 2

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1 and 2, plot the points in the Cartesian plane.

y

15.

16.

y

(2, 5)

1. 共⫺5, 3兲, 共1, ⫺1兲, 共⫺2, ⫺4兲, 共2, 0兲, 共1, ⫺6兲 2. 共0, ⫺4兲, 共5, 1兲, 共⫺3, 5兲, 共2, ⫺2兲, 共⫺6, ⫺1兲

(−3, 1)

c a

In Exercises 3–12, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. 3. 共3, 1兲, 共5, 5兲 5.



1 2,

1兲, 共

⫺ 32,

4. 共⫺3, 2兲, 共3, ⫺2兲

⫺5兲

6.

7. 共2, 2兲, 共4, 14兲

b

c x

x

a (2, − 2)

(6, − 2)

In Exercises 17–20, show that the points form the vertices of the given figure. (A rhombus is a quadrilateral whose sides have the same length.)

共23, ⫺ 13 兲, 共56, 1兲

8. 共⫺3, 7兲, 共1, ⫺1兲

9. 共1, 冪3 兲, 共⫺1, 1兲

(7, 4) b (7, 1)

Vertices

Figure

10. 共⫺2, 0兲, 共0,冪2 兲

17. 共0, 1兲, 共3, 7兲, 共4, ⫺1兲

Right triangle

11. 共0, ⫺4.8兲, 共0.5, 6兲

18. 共1, ⫺3兲, 共3, 2兲, 共⫺2, 4兲

Isosceles triangle

12. 共5.2, 6.4兲, 共⫺2.7, 1.8兲

19. 共0, 0兲, 共1, 2兲, 共2, 1兲, 共3, 3兲

Rhombus

20. 共0, 1兲, 共3, 7兲, 共4, 4兲, 共1, ⫺2兲

Parallelogram

In Exercises 13 – 16, (a) find the length of each side of the right triangle and (b) show that these lengths satisfy the Pythagorean Theorem. y

13.

14.

y

c (0, 0)

c a

b a

(4, 0) x

21. 共1, 0兲, 共x, ⫺4兲

(13, 6)

(4, 3)

(1, 1)

b x

(13, 1)

In Exercises 21 and 22, find x such that the distance between the points is 5. 22. 共2, ⫺1兲, 共x, 2兲

In Exercises 23 and 24, find y such that the distance between the points is 8. 23. 共0, 0兲, 共3, y兲

24. 共5, 1兲, 共5, y兲

The answers to the odd-numbered and selected even-numbered exercises are given in the back of the text. Worked-out solutions to the odd-numbered exercises are given in the Student Solutions Guide.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.1 25. Building Dimensions The base and height of the trusses for the roof of a house are 32 feet and 5 feet, respectively (see figure). (a) Find the distance from the eaves to the peak of the roof.

Dow Jones Industrial Average In Exercises 29 and 30, use the figure below showing the Dow Jones Industrial Average for common stocks. (Source: Dow Jones, Inc.) 29. Estimate the Dow Jones Industrial Average for each date.

(b) The length of the house is 40 feet. Use the result of part (a) to find the number of square feet of roofing.

c

(b) December 2005

(c) May 2006

(d) January 2007

200 ft

32 125 ft Figure for 25

(a) March 2005

30. Estimate the percent increase or decrease in the Dow Jones Industrial Average (a) from March 2005 to November 2005 and (b) from May 2006 to February 2007.

Figure for 26

26. Wire Length A guy wire is stretched from a broadcasting tower at a point 200 feet above the ground to an anchor 125 feet from the base (see figure). How long is the wire? In Exercises 27 and 28, use a graphing utility to graph a scatter plot, a bar graph, or a line graph to represent the data. Describe any trends that appear. 27. Consumer Trends The numbers (in millions) of basic cable television subscribers in the United States for 1996 through 2005 are shown in the table. (Source: National Cable & Telecommunications Association) Year

1996

1997

1998

1999

2000

Subscribers

62.3

63.6

64.7

65.5

66.3

Year

2001

2002

2003

2004

2005

Subscribers

66.7

66.5

66.0

65.7

65.3

Dow Jones Industrial Average

40

d

5

41

The Cartesian Plane and the Distance Formula

13,200 12,800 12,400 12,000 11,600 11,200 10,800 10,400 10,000

Jan. May Mar.

J F MAM J J A S O N D J F MAM J J A S O N D J F 2005 2006 2007

Figure for 29 and 30

Construction In Exercises 31 and 32, use the figure, which shows the median sales prices of existing onefamily homes sold (in thousands of dollars) in the United States from 1990 through 2005. (Source: National Association of Realtors) 31. Estimate the median sales price of existing one-family homes for each year. (a) 1990

Year

1996

1997

1998

1999

2000

Subscribers

44.0

55.3

69.2

86.0

109.5

Year

2001

2002

2003

2004

2005

Subscribers

128.4

140.8

158.7

182.1

207.9

(b) 1992

(c) 1997

(d) 2005

32. Estimate the percent increases in the value of existing one-family homes (a) from 1993 to 1994 and (b) from 2003 to 2004.

Median Sales Price (in thousands of dollars)

28. Consumer Trends The numbers (in millions) of cellular telephone subscribers in the United States for 1996 through 2005 are shown in the table. (Source: Cellular Telecommunications & Internet Association)

Dec.

220 200 180 160 140 120 100 80 60 1991 1993 1995 1997 1999 2001 2003 2005

Year Figure for 31 and 32

The symbol indicates an exercise in which you are instructed to use graphing technology or a symbolic computer algebra system. The solutions of other exercises may also be facilitated by use of appropriate technology.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Functions, Graphs, and Limits

Research Project In Exercises 33 and 34, (a) use the Midpoint Formula to estimate the revenue and profit of the company in 2003. (b) Then use your school’s library, the Internet, or some other reference source to find the actual revenue and profit for 2003. (c) Did the revenue and profit increase in a linear pattern from 2001 to 2005? Explain your reasoning. (d) What were the company’s expenses during each of the given years? (e) How would you rate the company’s growth from 2001 to 2005? (Source: The Walt Disney Company and CVS Corporation)

36. Health The percents of adults who are considered drinkers or smokers are shown in the table. Drinkers were defined as those who had five or more drinks in 1 day at least once during a recent year. Smokers were defined as those who smoked more than 100 cigarettes in their lifetime and smoked daily or semi-daily. (Source: National Health Interview Survey)

33. The Walt Disney Company Year

2001

Revenue (millions of $) Profit (millions of $)

2003

Year

2001

2002

2003

2004

2005

Drinkers

20.0

19.9

19.1

19.1

19.5

Smokers

22.7

22.4

21.6

20.9

20.9

2005

(a) Sketch a line graph of each data set.

25,269

31,944

(b) Describe any trends that appear.

2058.0

2729.0

34. CVS Corporation Year

2001

2003

2005

Revenue (millions of $)

22,241

37,006

Profit (millions of $)

638.0

1172.1

35. Economics The table shows the numbers of ear infections treated by doctors at HMO clinics of three different sizes: small, medium, and large. Number of doctors

0

1

2

3

4

Cases per small clinic

0

20

28

35

40

Cases per medium clinic

0

30

42

53

60

Cases per large clinic

0

35

49

62

70

(a) Show the relationship between doctors and treated ear infections using three curves, where the number of doctors is on the horizontal axis and the number of ear infections treated is on the vertical axis. (b) Compare the three relationships. (Source: Adapted from Taylor, Economics, Fifth Edition)

Computer Graphics In Exercises 37 and 38, the red figure is translated to a new position in the plane to form the blue figure. (a) Find the vertices of the transformed figure. (b) Then use a graphing utility to draw both figures. y

37.

38.

3 1

(−3, −1)

(1, 3)

3

(0, 2)

(3, 1)

3 units 1 x

1

2

(0, 0) (−1, − 2)

−3

1

3

(2, 0)

2 units

3

x

39. Use the Midpoint Formula repeatedly to find the three points that divide the segment joining 共x1, y1兲 and 共x2, y2兲 into four equal parts. 40. Show that 共13 关2x1 ⫹ x2 兴, 13 关2y1 ⫹ y2 兴 兲 is one of the points of trisection of the line segment joining 共x1, y1兲 and 共x2, y2兲. Then, find the second point of trisection by finding the midpoint of the segment joining

冢13 关2x

1



1 ⫹ x2 兴, 关2y1 ⫹ y2 兴 and 共x2, y2 兲. 3

41. Use Exercise 39 to find the points that divide the line segment joining the given points into four equal parts. (a) 共1, ⫺2兲, 共4, ⫺1兲

(b) 共⫺2, ⫺3兲, 共0, 0兲

42. Use Exercise 40 to find the points of trisection of the line segment joining the given points. (a) 共1, ⫺2兲, 共4, 1兲

The symbol

y

3 units

CHAPTER 1

3 units

42

(b) 共⫺2, ⫺3兲, 共0, 0兲

indicates an exercise that contains material from textbooks in other disciplines.

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SECTION 1.2

Graphs of Equations

43

Section 1.2 ■ Sketch graphs of equations by hand.

Graphs of Equations

■ Find the x- and y-intercepts of graphs of equations. ■ Write the standard forms of equations of circles. ■ Find the points of intersection of two graphs. ■ Use mathematical models to model and solve real-life problems.

The Graph of an Equation In Section 1.1, you used a coordinate system to represent graphically the relationship between two quantities. There, the graphical picture consisted of a collection of points in a coordinate plane (see Example 2 in Section 1.1). Frequently, a relationship between two quantities is expressed as an equation. For instance, degrees on the Fahrenheit scale are related to degrees on the Celsius scale by the equation F ⫽ 95C ⫹ 32. In this section, you will study some basic procedures for sketching the graphs of such equations. The graph of an equation is the set of all points that are solutions of the equation.

Example 1 y 8 6

Sketch the graph of y ⫽ 7 ⫺ 3x. SOLUTION The simplest way to sketch the graph of an equation is the pointplotting method. With this method, you construct a table of values that consists of several solution points of the equation, as shown in the table below. For instance, when x ⫽ 0

(0, 7)

4

(1, 4)

2 −6 −4 −2

Sketching the Graph of an Equation

−2

(2, 1) 2

4

(3, − 2)

−4

8

x

y ⫽ 7 ⫺ 3共0兲 ⫽ 7 which implies that 共0, 7兲 is a solution point of the graph.

(4, − 5)

−6

FIGURE 1.13 y ⫽ 7 ⫺ 3x

6

Solution Points for

✓CHECKPOINT 1 Sketch the graph of y ⫽ 2x ⫹ 1.



x

0

1

2

3

4

y ⫽ 7 ⫺ 3x

7

4

1

⫺2

⫺5

From the table, it follows that 共0, 7兲, 共1, 4兲, 共2, 1兲, 共3, ⫺2兲, and 共4, ⫺5兲 are solution points of the equation. After plotting these points, you can see that they appear to lie on a line, as shown in Figure 1.13. The graph of the equation is the line that passes through the five plotted points.

STUDY TIP Even though we refer to the sketch shown in Figure 1.13 as the graph of y ⫽ 7 ⫺ 3x, it actually represents only a portion of the graph. The entire graph is a line that would extend off the page.

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44

CHAPTER 1

Functions, Graphs, and Limits

Example 2 STUDY TIP The graph shown in Example 2 is a parabola. The graph of any second-degree equation of the form y ⫽ ax 2 ⫹ bx ⫹ c, a ⫽ 0 has a similar shape. If a > 0, the parabola opens upward, as in Figure 1.14(b), and if a < 0, the parabola opens downward.

Sketching the Graph of an Equation

Sketch the graph of y ⫽ x 2 ⫺ 2. Begin by constructing a table of values, as shown below.

SOLUTION

x

⫺2

⫺1

0

1

2

3

y ⫽ x2 ⫺ 2

2

⫺1

⫺2

⫺1

2

7

Next, plot the points given in the table, as shown in Figure 1.14(a). Finally, connect the points with a smooth curve, as shown in Figure 1.14(b). y

y

8

8

(3, 7)

(−2, 2)

−4

6

6

4

4

(2, 2)

2

−2

(−1, − 1)

(1, − 1) 4

y = x2 − 2

2

6

x

−4

−2

2

4

6

x

−2

(0, − 2)

(a)

(b)

FIGURE 1.14

✓CHECKPOINT 2 Sketch the graph of y ⫽ x2 ⫺ 4. y

x



The point-plotting technique demonstrated in Examples 1 and 2 is easy to use, but it does have some shortcomings. With too few solution points, you can badly misrepresent the graph of a given equation. For instance, how would you connect the four points in Figure 1.15? Without further information, any one of the three graphs in Figure 1.16 would be reasonable. y

y

y

FIGURE 1.15 x

x

x

FIGURE 1.16

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SECTION 1.2

45

Graphs of Equations

Intercepts of a Graph

Algebra Review

It is often easy to determine the solution points that have zero as either the x-coordinate or the y-coordinate. These points are called intercepts because they are the points at which the graph intersects the x- or y-axis. Some texts denote the x-intercept as the x-coordinate of the point 共a, 0兲 rather than the point itself. Unless it is necessary to make a distinction, we will use the term intercept to mean either the point or the coordinate. A graph may have no intercepts or several intercepts, as shown in Figure 1.17.

Finding intercepts involves solving equations. For a review of some techniques for solving equations, see page 106.

y

y

y

x

y

x

x

Three x-intercepts One y-intercept

No x-intercept One y-intercept

One x-intercept Two y-intercepts

x

No intercepts

FIGURE 1.17

Finding Intercepts

y

1. To find x-intercepts, let y be zero and solve the equation for x.

y = x 3 − 4x 4

2. To find y-intercepts, let x be zero and solve the equation for y.

3

(−2, 0) −4 −3

(0, 0) −1 −1

(2, 0)

1

3

4

−3

Finding x- and y-Intercepts

Find the x- and y-intercepts of the graph of each equation. b. x ⫽ y 2 ⫺ 3

SOLUTION

−4

a. Let y ⫽ 0. Then 0 ⫽ x共x 2 ⫺ 4兲 ⫽ x共x ⫹ 2兲共x ⫺ 2兲 has solutions x ⫽ 0 and x ⫽ ± 2. Let x ⫽ 0. Then y ⫽ 共0兲3 ⫺ 4共0兲 ⫽ 0.

FIGURE 1.18

x-intercepts: 共0, 0兲, 共2, 0兲, 共⫺2, 0兲

y

y-intercept: 共0, 0兲

b. Let y ⫽ 0. Then x ⫽ 共0兲 ⫺ 3 ⫽ ⫺3. Let x ⫽ 0. Then solutions y ⫽ ± 冪3. 2

x = y2 − 3

2 1

(0,

3)

x-intercept: 共⫺3, 0兲

(−3, 0) −2

Example 3

a. y ⫽ x 3 ⫺ 4x

−2

−4

x

−1

1 −1 −2

FIGURE 1.19

x

(0, −

3)

y-intercepts: 共0, 冪3 兲, 共0, ⫺冪3 兲

See Figure 1.18.

y2

⫺ 3 ⫽ 0 has See Figure 1.19.

✓CHECKPOINT 3 Find the x- and y-intercepts of the graph of each equation. a. y ⫽ x2 ⫺ 2x ⫺ 3

b. y2 ⫺ 4 ⫽ x



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46

CHAPTER 1

Functions, Graphs, and Limits

TECHNOLOGY

Zooming in to Find Intercepts You can use the zoom feature of a graphing utility to approximate the x-intercepts of a graph. Suppose you want to approximate the x-intercept(s) of the graph of y ⫽ 2x3 ⫺ 3x ⫹ 2. STUDY TIP Some graphing utilities have a built-in program that can find the x-intercepts of a graph. If your graphing utility has this feature, try using it to find the x-intercept of the graph shown on the left. (Your calculator may call this the root or zero feature.)*

Begin by graphing the equation, as shown below in part (a). From the viewing window shown, the graph appears to have only one x-intercept. This intercept lies between ⫺2 and ⫺1. By zooming in on the intercept, you can improve the approximation, as shown in part (b). To three decimal places, the solution is x ⬇ ⫺1.476. y = 2x 3 − 3x + 2

y = 2x 3 − 3x + 2

4

−6

0.1

6

−1.48

−1.47

−4

−0.1

(a)

(b)

Here are some suggestions for using the zoom feature. 1. With each successive zoom-in, adjust the x-scale so that the viewing window shows at least one tick mark on each side of the x-intercept. 2. The error in your approximation will be less than the distance between two scale marks. 3. The trace feature can usually be used to add one more decimal place of accuracy without changing the viewing window. Part (a) below shows the graph of y ⫽ x 2 ⫺ 5x ⫹ 3. Parts (b) and (c) show “zoom-in views” of the two intercepts. From these views, you can approximate the x-intercepts to be x ⬇ 0.697 and x ⬇ 4.303. 0.01

10

10

−10

− 10

(a)

y = x 2 − 5x + 3

0.01

0.68

− 0.01

(b)

0.71

y = x 2 − 5x + 3

4.29

−0.01

4.32

y = x 2 − 5x + 3

(c)

*Specific calculator keystroke instructions for operations in this and other technology boxes can be found at college.hmco.com/info/larsonapplied.

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SECTION 1.2

Graphs of Equations

47

Circles y

Throughout this course, you will learn to recognize several types of graphs from their equations. For instance, you should recognize that the graph of a seconddegree equation of the form y ⫽ ax 2 ⫹ bx ⫹ c, a ⫽ 0

Center: (h, k)

is a parabola (see Example 2). Another easily recognized graph is that of a circle. Consider the circle shown in Figure 1.20. A point 共x, y兲 is on the circle if and only if its distance from the center 共h, k兲 is r. By the Distance Formula,

Radius: r Point on circle: (x, y) x

FIGURE 1.20

冪共x ⫺ h兲 2 ⫹ 共 y ⫺ k兲 2 ⫽ r.

By squaring both sides of this equation, you obtain the standard form of the equation of a circle. Standard Form of the Equation of a Circle

The point 共x, y兲 lies on the circle of radius r and center 共h, k兲 if and only if

共x ⫺ h兲2 ⫹ 共 y ⫺ k兲2 ⫽ r 2. From this result, you can see that the standard form of the equation of a circle with its center at the origin, 共h, k兲 ⫽ 共0, 0兲, is simply x 2 ⫹ y 2 ⫽ r 2.

Example 4

y

6

SOLUTION (3, 4)

(−1, 2) −6

−2

4 −2 −4

(x + 1)2 + (y − 2)2 = 20

FIGURE 1.21

Finding the Equation of a Circle

The point 共3, 4兲 lies on a circle whose center is at 共⫺1, 2兲, as shown in Figure 1.21. Find the standard form of the equation of this circle.

8

4

Circle with center at origin

x

The radius of the circle is the distance between 共⫺1, 2兲 and 共3, 4兲.

r ⫽ 冪 关3 ⫺ 共⫺1兲兴 2 ⫹ 共4 ⫺ 2兲 2 ⫽ 冪16 ⫹ 4 ⫽ 冪20

Distance Formula Simplify. Radius

Using 共h, k兲 ⫽ 共⫺1, 2兲, the standard form of the equation of the circle is

共x ⫺ h兲 2 ⫹ 共 y ⫺ k兲 2 ⫽ r 2 2 关x ⫺ 共⫺1兲兴 2 ⫹ 共 y ⫺ 2兲2 ⫽ 共冪20 兲 共x ⫹ 1兲 2 ⫹ 共 y ⫺ 2兲 2 ⫽ 20.

Substitute for h, k, and r. Write in standard form.

✓CHECKPOINT 4 The point 共1, 5兲 lies on a circle whose center is at 共⫺2, 1兲. Find the standard form of the equation of this circle. ■

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48

CHAPTER 1

Functions, Graphs, and Limits

TECHNOLOGY

General Form of the Equation of a Circle

To graph a circle on a graphing utility, you can solve its equation for y and graph the top and bottom halves of the circle separately. For instance, you can graph the circle 共x ⫹ 1兲 2 ⫹ 共 y ⫺ 2兲 2 ⫽ 20 by graphing the following equations. y ⫽ 2 ⫹ 冪20 ⫺ 共x ⫹ 1兲

Ax 2 ⫹ Ay 2 ⫹ Dx ⫹ Ey ⫹ F ⫽ 0, A ⫽ 0 To change from general form to standard form, you can use a process called completing the square, as demonstrated in Example 5.

Example 5

Sketch the graph of the circle whose general equation is

2

y ⫽ 2 ⫺ 冪20 ⫺ 共x ⫹ 1兲 2

4x 2 ⫹ 4y 2 ⫹ 20x ⫺ 16y ⫹ 37 ⫽ 0. SOLUTION

First divide by 4 so that the coefficients of x 2 and y 2 are both 1.

4x 2 ⫹ 4y 2 ⫹ 20x ⫺ 16y ⫹ 37 ⫽ 0 x 2 ⫹ y 2 ⫹ 5x ⫺ 4y ⫹ 37 4 ⫽ 0 2 2 共x ⫹ 5x ⫹ 䊏兲 ⫹ 共 y ⫺ 4y ⫹ 䊏兲 ⫽ ⫺ 374 共x 2 ⫹ 5x ⫹ 254 兲 ⫹ 共 y 2 ⫺ 4y ⫹ 4兲 ⫽ ⫺ 374 ⫹

If you want the result to appear circular, you need to use a square setting, as shown below. 10

− 10

Completing the Square

10

共Half兲 2

Write original equation. Divide each side by 4. Group terms. 25 4

⫹4

Complete the square.

共Half兲 2

共x ⫹ 52 兲2 ⫹ 共 y ⫺ 2兲2 ⫽ 1

Write in standard form.

From the standard form, you can see that the circle is centered at 共⫺ 52, 2兲 and has a radius of 1, as shown in Figure 1.22.

− 10

Standard setting 9

(x + 52 ( + (y − 2) 2

2

y

=1 3

r=1 2

(− 52 , 2(

9

−9

1

−3

Square setting −4

−3

−2

−1

x

FIGURE 1.22

The general equation Ax 2 ⫹ Ay 2 ⫹ Dx ⫹ Ey ⫹ F ⫽ 0 may not always represent a circle. In fact, such an equation will have no solution points if the procedure of completing the square yields the impossible result

共x ⫺ h兲 2 ⫹ 共 y ⫺ k兲 2 ⫽ negative number.

No solution points

✓CHECKPOINT 5 Write the equation of the circle x 2 ⫹ y 2 ⫺ 4x ⫹ 2y ⫹ 1 ⫽ 0 in standard form and sketch its graph. ■

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SECTION 1.2

Graphs of Equations

49

Points of Intersection A point of intersection of two graphs is an ordered pair that is a solution point of both graphs. For instance, Figure 1.23 shows that the graphs of

4

−6

y ⫽ x2 ⫺ 3

6

have two points of intersection: 共2, 1兲 and 共⫺1, ⫺2兲. To find the points analytically, set the two y-values equal to each other and solve the equation x2 ⫺ 3 ⫽ x ⫺ 1

−4

FIGURE 1.23

STUDY TIP The Technology note on page 46 describes how to use a graphing utility to find the x-intercepts of a graph. A similar procedure can be used to find the points of intersection of two graphs. (Your calculator may call this the intersect feature.)

for x. A common business application that involves points of intersection is breakeven analysis. The marketing of a new product typically requires an initial investment. When sufficient units have been sold so that the total revenue has offset the total cost, the sale of the product has reached the break-even point. The total cost of producing x units of a product is denoted by C, and the total revenue from the sale of x units of the product is denoted by R. So, you can find the break-even point by setting the cost C equal to the revenue R, and solving for x.

Example 6 MAKE A DECISION

Cost equation

The total revenue from the sale of x units is given by

y

R ⫽ 1.2x.

50,000 45,000

Revenue equation

To find the break-even point, set the cost equal to the revenue and solve for x.

40,000

Sales (in dollars)

The total cost of producing x units of the product is given by

C ⫽ 0.65x ⫹ 10,000.

Break-Even Analysis

R⫽C 1.2x ⫽ 0.65x ⫹ 10,000 0.55x ⫽ 10,000

Break-even point: 18,182 units

35,000

C = 0.65x + 10,000

10,000 0.55 x ⬇ 18,182

25,000 20,000

Finding a Break-Even Point

A business manufactures a product at a cost of $0.65 per unit and sells the product for $1.20 per unit. The company’s initial investment to produce the product was $10,000. Will the company break even if it sells 18,000 units? How many units must the company sell to break even? SOLUTION

30,000

y⫽x⫺1

and

x⫽

Profit

Loss

15,000

R = 1.2x 10,000

20,000

Number of units

FIGURE 1.24

Substitute for R and C. Subtract 0.65x from each side. Divide each side by 0.55. Use a calculator.

No, the company will not break even if it sells 18,000 units. The company must sell 18,182 units before it breaks even. This result is shown graphically in Figure 1.24.

10,000 5,000

Set revenue equal to cost.

x

✓CHECKPOINT 6 How many units must the company in Example 6 sell to break even if the selling price is $1.45 per unit? ■

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50

CHAPTER 1

Functions, Graphs, and Limits

p

Two types of applications that economists use to analyze a market are supply and demand equations. A supply equation shows the relationship between the unit price p of a product and the quantity supplied x. The graph of a supply equation is called a supply curve. (See Figure 1.25.) A typical supply curve rises because producers of a product want to sell more units if the unit price is higher. A demand equation shows the relationship between the unit price p of a product and the quantity demanded x. The graph of a demand equation is called a demand curve. (See Figure 1.26.) A typical demand curve tends to show a decrease in the quantity demanded with each increase in price. In an ideal situation, with no other factors present to influence the market, the production level should stabilize at the point of intersection of the graphs of the supply and demand equations. This point is called the equilibrium point. The x-coordinate of the equilibrium point is called the equilibrium quantity and the p-coordinate is called the equilibrium price. (See Figure 1.27.) You can find the equilibrium point by setting the demand equation equal to the supply equation and solving for x.

x

FIGURE 1.25

Supply Curve

p

x

FIGURE 1.26

Demand Curve

p

Demand

Equilibrium p0 price

Example 7

The demand and supply equations for a DVD player are given by p ⫽ 195 ⫺ 5.8x p ⫽ 150 ⫹ 3.2x

Supply

Equilibrium point (x0, p0) x

x0 Equilibrium quantity

FIGURE 1.27

Equilibrium Point

Equilibrium Point

Price per unit (in dollars)

250 (5, 166) Supply

150

Demand equation Supply equation

where p is the price in dollars and x represents the number of units in millions. Find the equilibrium point for this market. SOLUTION

Begin by setting the demand equation equal to the supply equation.

195 ⫺ 5.8x ⫽ 150 ⫹ 3.2x 45 ⫺ 5.8x ⫽ 3.2x 45 ⫽ 9x 5⫽x

Set equations equal to each other. Subtract 150 from each side. Add 5.8x to each side. Divide each side by 9.

So, the equilibrium point occurs when the demand and supply are each five million units. (See Figure 1.28.) The price that corresponds to this x-value is obtained by substituting x ⫽ 5 into either of the original equations. For instance, substituting into the demand equation produces

p

200

Finding the Equilibrium Point

p ⫽ 195 ⫺ 5.8共5兲 ⫽ 195 ⫺ 29 ⫽ $166. Substitute x ⫽ 5 into the supply equation to see that you obtain the same price.

Demand

100

✓CHECKPOINT 7

50 1 2 3 4 5 6 7 8 9

Number of units (in millions)

FIGURE 1.28

x

The demand and supply equations for a calculator are p ⫽ 136 ⫺ 3.5x and p ⫽ 112 ⫹ 2.5x, respectively, where p is the price in dollars and x represents the number of units in millions. Find the equilibrium point for this market. ■

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.2

51

Graphs of Equations

Mathematical Models In this text, you will see many examples of the use of equations as mathematical models of real-life phenomena. In developing a mathematical model to represent actual data, you should strive for two (often conflicting) goals— accuracy and simplicity.

Example 8

Algebra Review For help in evaluating the expressions in Example 8, see the review of order of operations on page 105.

Using Mathematical Models

The table shows the annual sales (in millions of dollars) for Dillard’s and Kohl’s for 2001 to 2005. In the summer of 2006, the publication Value Line listed the projected 2006 sales for the companies as $7625 million and $15,400 million, respectively. How do you think these projections were obtained? (Source: Dillard’s Inc. and Kohl’s Corp.)

Year

2001

2002

2003

2004

2005

t

1

2

3

4

5

Dillard’s

8155

7911

7599

7529

7560

Kohl’s

7489

9120

10,282

11,701

13,402

Annual Sales S

The projections were obtained by using past sales to predict future sales. The past sales were modeled by equations that were found by a statistical procedure called least squares regression analysis. SOLUTION

Annual sales (in millions of dollars)

14,000 13,000 12,000

S ⫽ 56.57t 2 ⫺ 496.6t ⫹ 8618, 1 ≤ t ≤ 5 S ⫽ 28.36t2 ⫹ 1270.6t ⫹ 6275, 1 ≤ t ≤ 5

Kohl's

11,000 10,000

Dillard's

9,000

Dillard’s Kohl’s

Using t ⫽ 6 to represent 2006, you can predict the 2006 sales to be

8,000 7,000 1

2

3

4

Year (1 ↔ 2001)

FIGURE 1.29

5

S ⫽ 56.57共6兲2 ⫺ 496.6共6兲 ⫹ 8618 ⬇ 7675 S ⫽ 28.36共6兲2 ⫹ 1270.6共6兲 ⫹ 6275 ⬇ 14,920

t

Dillard’s Kohl’s

These two projections are close to those projected by Value Line. The graphs of the two models are shown in Figure 1.29.

✓CHECKPOINT 8 The table shows the annual sales (in millions of dollars) for Dollar General for 1999 through 2005. In the summer of 2005, the publication Value Line listed projected 2006 sales for Dollar General as $9300 million. How does this projection compare with the projection obtained using the model below? (Source: Dollar General Corp.)

S ⫽ 16.246t 2 ⫹ 390.53t ⫺ 951.2,

9 ≤ t ≤ 15

Year

1999

2000

2001

2002

2003

2004

2005

t

9

10

11

12

13

14

15

Sales

3888.0

4550.6

5322.9

6100.4

6872.0

7660.9

8582.2 ■

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52

CHAPTER 1

Functions, Graphs, and Limits

STUDY TIP To test the accuracy of a model, you can compare the actual data with the values given by the model. For instance, the table below compares the actual Kohl’s sales with those given by the model found in Example 8. Year

2001

2002

2003

2004

2005

Actual

7489

9120

10,282

11,701

13,402

Model

7574.0

8929.6

10,342.0

11,811.2

13,337.0

Much of your study of calculus will center around the behavior of the graphs of mathematical models. Figure 1.30 shows the graphs of six basic algebraic equations. Familiarity with these graphs will help you in the creation and use of mathematical models. y

y

y

y=x

−2

2

4

2

1

3

1

−1

1

2

x

2

−1

y=

1

−2

−2

−1

1

2

−2

x2

−1

1

x

−2

(c) Cubic model

y y

y

3

y=

2

x

y = 1x

2

4

3

x

2

−1

(b) Quadratic model

(a) Linear model

y = x3

1

y = ⏐x⏐ −1

2

−1

1

2

x

1

1

1

2

3

(d) Square root model

x

−2

−1

1

2

x

(e) Absolute value model

(f) Rational model

FIGURE 1.30

CONCEPT CHECK 1. What does the graph of an equation represent? 2. Describe how to find the x- and y-intercepts of the graph of an equation. 3. How can you check that an ordered pair is a point of intersection of two graphs? 4. Can the graph of an equation have more than one y-intercept?

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.2

53

Graphs of Equations

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 0.3 and 0.4.

Skills Review 1.2

In Exercises 1–6, solve for y. 1. 5y ⫺ 12 ⫽ x

2. ⫺y ⫽ 15 ⫺ x

3. x 3 y ⫹ 2y ⫽ 1

4. x 2 ⫹ x ⫺ y 2 ⫺ 6 ⫽ 0

5. 共x ⫺ 2兲 ⫹ 共 y ⫹ 1兲 ⫽ 9 2

6. 共x ⫹ 6兲 2 ⫹ 共 y ⫺ 5兲 2 ⫽ 81

2

In Exercises 7–10, complete the square to write the expression as a perfect square trinomial. 7. x 2 ⫺ 4x ⫹ 䊏 9.

x2

⫺ 5x ⫹

䊏 ⫹ 3x ⫹ 䊏

8. x 2 ⫹ 6x ⫹



10.

x2

In Exercises 11–14, factor the expression. 11. x 2 ⫺ 3x ⫹ 2 13. y 2 ⫺ 3y ⫹

12. x 2 ⫹ 5x ⫹ 6

9 4

14. y 2 ⫺ 7y ⫹

Exercises 1.2

49 4

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 4, determine whether the points are solution points of the given equation.

y

(c)

1. 2x ⫺ y ⫺ 3 ⫽ 0 (a) 共1, 2兲

(b) 共1, ⫺1兲

(c) 共4, 5兲

(a) 共6, ⫺9兲

(b) 共⫺5, 10兲

(c)

共12 , 58 兲

(c)

共32, 72 兲

3

2

2

1

1

2. 7x ⫹ 4y ⫺ 6 ⫽ 0 −3

y

(d)

−1

−2

x

1

4.

x2y



(a) 共0,

x2 1 5

(b)

共12, ⫺1兲



(b) 共2, 4兲

5. y ⫽ x ⫺ 2

6. y ⫽

7. y ⫽

8. y ⫽ 冪9 ⫺ x 2

⫹ 2x

ⱍⱍ

9. y ⫽ x ⫺ 2 (a)

1

3

1 −2

−1 −2

1

2

x

−1

2 1

−2 −3 −2 −1

2

3

2

11. 2x ⫺ y ⫺ 3 ⫽ 0

12. 4x ⫺ 2y ⫺ 5 ⫽ 0

13. y ⫽ x 2 ⫹ x ⫺ 2

14. y ⫽ x 2 ⫺ 4x ⫹ 3

15. y ⫽ 冪4 ⫺

1 1

1

3

x

In Exercises 11–20, find the x- and y-intercepts of the graph of the equation.

y

(b)

4

2

⫹2

10. y ⫽ x 3 ⫺ x

y

5

x

(c) 共⫺2, ⫺4兲

⫺ 12x

y

(f)

⫺ 5y ⫽ 0

In Exercises 5–10, match the equation with its graph. Use a graphing utility, set for a square setting, to confirm your result. [The graphs are labeled (a)–(f).] x2

y

(e)

−1

x

2

−2

3. x 2 ⫹ y 2 ⫽ 4 (a) 共1, ⫺冪3 兲

1

4

x

17. y ⫽

x2

x2 ⫺ 4 x⫺2

19. x 2 y ⫺ x 2 ⫹ 4y ⫽ 0

16. y 2 ⫽ x 3 ⫺ 4x 18. y ⫽

x 2 ⫹ 3x 2x

20. 2x 2 y ⫹ 8y ⫺ x 2 ⫽ 1

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54

CHAPTER 1

Functions, Graphs, and Limits

In Exercises 21–36, sketch the graph of the equation and label the intercepts. Use a graphing utility to verify your results. 21. y ⫽ 2x ⫹ 3

22. y ⫽ ⫺3x ⫹ 2

23. y ⫽ x ⫺ 3

24. y ⫽ x ⫹ 6

2

25. y ⫽ 共x ⫺ 1兲

2

2

28. y ⫽ 1 ⫺ x 3

29. y ⫽ ⫺ 冪x ⫺ 1

30. y ⫽ 冪x ⫹ 1

31. y ⫽ x ⫹ 1



32. y ⫽ ⫺ x ⫺ 2

33. y ⫽ 1兾共x ⫺ 3兲

34. y ⫽ 1兾共x ⫹ 1)

35. x ⫽ y 2 ⫺ 4

36. x ⫽ 4 ⫺ y 2



(c) How many units would yield a profit of $1000?



In Exercises 37–44, write the general form of the equation of the circle. 37. Center: 共0, 0兲; radius: 4

(a) Find equations for the total cost C and total revenue R for x units. (b) Find the break-even point by finding the point of intersection of the cost and revenue equations.

26. y ⫽ 共5 ⫺ x兲 2

27. y ⫽ x 3 ⫹ 2



61. Break-Even Analysis You are setting up a part-time business with an initial investment of $15,000. The unit cost of the product is $11.80, and the selling price is $19.30.

38. Center: 共0, 0兲; radius: 5

39. Center: 共2, ⫺1兲; radius: 3 40. Center: 共⫺4, 3兲; radius: 2 41. Center: 共⫺1, 2兲; solution point: 共0, 0兲 42. Center: 共3, ⫺2兲; solution point: 共⫺1, 1兲 43. Endpoints of a diameter: 共0, 0兲, 共6, 8兲 44. Endpoints of a diameter: 共⫺4, ⫺1兲, 共4, 1兲 In Exercises 45–52, complete the square to write the equation of the circle in standard form. Then use a graphing utility to graph the circle.

62. Break-Even Analysis A 2006 Chevrolet Impala costs $23,730 with a gasoline engine. A 2006 Toyota Camry costs $25,900 with a hybrid engine. The Impala gets 20 miles per gallon of gasoline and the Camry gets 34 miles per gallon of gasoline. Assume that the price of gasoline is $2.909. (Source: Consumer Reports, March and August 2006) (a) Show that the cost Cg of driving the Chevrolet Impala x miles is Cg ⫽ 23,730 ⫹ 2.909x兾20 and the cost Ch of driving the Toyota Camry x miles is Ch ⫽ 25,900 ⫹ 2.909x兾34 (b) Find the break-even point. That is, find the mileage at which the hybrid-powered Toyota Camry becomes more economical than the gasoline-powered Chevrolet Impala.

46. x 2 ⫹ y 2 ⫺ 2x ⫹ 6y ⫺ 15 ⫽ 0

Break-Even Analysis In Exercises 63–66, find the sales necessary to break even for the given cost and revenue equations. (Round your answer up to the nearest whole unit.) Use a graphing utility to graph the equations and then find the break-even point.

47. x2 ⫹ y2 ⫺ 4x ⫺ 2y ⫹ 1 ⫽ 0

63. C ⫽ 0.85x ⫹ 35,000, R ⫽ 1.55x

45. x 2 ⫹ y 2 ⫺ 2x ⫹ 6y ⫹ 6 ⫽ 0

48. x 2 ⫹ y 2 ⫺ 4x ⫹ 2y ⫹ 3 ⫽ 0 49. 2x 2 ⫹ 2y 2 ⫺ 2x ⫺ 2y ⫺ 3 ⫽ 0 50. 4x 2 ⫹ 4y 2 ⫺ 4x ⫹ 2y ⫺ 1 ⫽ 0 51. 4x2 ⫹ 4y2 ⫹ 12x ⫺ 24y ⫹ 41 ⫽ 0 52. 3x 2 ⫹ 3y 2 ⫺ 6y ⫺ 1 ⫽ 0 In Exercises 53–60, find the points of intersection (if any) of the graphs of the equations. Use a graphing utility to check your results. 53. x ⫹ y ⫽ 2, 2x ⫺ y ⫽ 1 54. x ⫹ y ⫽ 7, 3x ⫺ 2y ⫽ 11 55. x 2 ⫹ y 2 ⫽ 25, 2x ⫹ y ⫽ 10 56. x2 ⫺ y ⫽ ⫺2, x ⫺ y ⫽ ⫺4 57. y ⫽ x 3, y ⫽ 2x 58. y ⫽ 冪x, y ⫽ x 59. y ⫽ x 4 ⫺ 2x 2 ⫹ 1, y ⫽ 1 ⫺ x 2 60. y ⫽ x 3 ⫺ 2x 2 ⫹ x ⫺ 1, y ⫽ ⫺x 2 ⫹ 3x ⫺ 1

64. C ⫽ 6x ⫹ 500,000, R ⫽ 35x 65. C ⫽ 8650x ⫹ 250,000, R ⫽ 9950x 66. C ⫽ 2.5x ⫹ 10,000, R ⫽ 4.9x 67. Supply and Demand The demand and supply equations for an electronic organizer are given by p ⫽ 180 ⫺ 4x

Demand equation

p ⫽ 75 ⫹ 3x

Supply equation

where p is the price in dollars and x represents the number of units, in thousands. Find the equilibrium point for this market. 68. Supply and Demand The demand and supply equations for a portable CD player are given by p ⫽ 190 ⫺ 15x

Demand equation

p ⫽ 75 ⫹ 8x

Supply equation

where p is the price in dollars and x represents the number of units, in hundreds of thousands. Find the equilibrium point for this market.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.2 69. Textbook Spending The amounts of money y (in millions of dollars) spent on college textbooks in the United States in the years 2000 through 2005 are shown in the table. (Source: Book Industry Study Group, Inc.) Year

2000

2001

2002

2003

2004

2005

Expense

4265

4571

4899

5086

5479

5703

A mathematical model for the data is given by y ⫽ 0.796t 3 ⫺ 8.65t 2 ⫹ 312.9t ⫹ 4268, where t represents the year, with t ⫽ 0 corresponding to 2000. (a) Compare the actual expenses with those given by the model. How well does the model fit the data? Explain your reasoning. (b) Use the model to predict the expenses in 2013. 70. Federal Education Spending The federal outlays y (in billions of dollars) for elementary, secondary, and vocational education in the United States in the years 2001 through 2005 are shown in the table. (Source: U.S. Office of Management and Budget) Year

2000

2001

2002

2003

2004

2005

Outlay

20.6

22.9

25.9

31.5

34.4

38.3

A mathematical model for the data is given by y ⫽ 0.136t ⫹ 3.00t ⫹ 20.2 2

where t represents the year, with t ⫽ 0 corresponding to 2000. (a) Compare the actual outlays with those given by the model. How well does the model fit the data? Explain. (b) Use the model to predict the outlays in 2012. 71. MAKE A DECISION: WEEKLY SALARY A mathematical model for the average weekly salary y of a person in finance and insurance is given by the equation y ⫽ ⫺0.77t 2 ⫹ 27.3t ⫹ 587, where t represents the year, with t ⫽ 0 corresponding to 2000. (Source: U.S. Bureau of Labor Statistics) (a) Use the model to complete the table. Year

2000

2001

2002

2003

(c) What does this model predict the average weekly salary to be in 2009? Do you think this prediction is valid? 72. MAKE A DECISION: KIDNEY TRANSPLANTS A mathematical model for the numbers of kidney transplants performed in the United States in the years 2001 through 2005 is given by y ⫽ 30.57t 2 ⫹ 381.4t ⫹ 13,852, where y is the number of transplants and t represents the year, with t ⫽ 1 corresponding to 2001. (Source: United Network for Organ Sharing) (a) Use a graphing utility or a spreadsheet to complete the table. Year

2001

2002

2003

2004

2005

Transplants (b) Use your school’s library, the Internet, or some other reference source to find the actual numbers of kidney transplants for the years 2001 through 2005. Compare the actual numbers with those given by the model. How well does the model fit the data? Explain your reasoning. (c) Using this model, what is the prediction for the number of transplants in the year 2011? Do you think this prediction is valid? What factors could affect this model’s accuracy? 73. Use a graphing utility to graph the equation y ⫽ cx ⫹ 1 for c ⫽ 1, 2, 3, 4, and 5. Then make a conjecture about the x-coefficient and the graph of the equation. 74. Break-Even Point Define the break-even point for a business marketing a new product. Give examples of a linear cost equation and a linear revenue equation for which the break-even point is 10,000 units. In Exercises 75–80, use a graphing utility to graph the equation and approximate the x- and y-intercepts of the graph. 75. y ⫽ 0.24x 2 ⫹ 1.32x ⫹ 5.36 76. y ⫽ ⫺0.56x 2 ⫺ 5.34x ⫹ 6.25 77. y ⫽ 冪0.3x 2 ⫺ 4.3x ⫹ 5.7 78. y ⫽ 冪⫺1.21x 2 ⫹ 2.34x ⫹ 5.6

2004

2007

Salary (b) This model was created using actual data from 2000 through 2005. How accurate do you think the model is in predicting the 2007 average weekly salary? Explain your reasoning.

The symbol

55

Graphs of Equations

79. y ⫽

0.2x 2 ⫹ 1 0.1x ⫹ 2.4

80. y ⫽

0.4x ⫺ 5.3 0.4x 2 ⫹ 5.3

81. Extended Application To work an extended application analyzing the numbers of workers in the farm work force in the United States from 1955 through 2005, visit this text’s website at college.hmco.com. (Data Source: U.S. Bureau of Labor Statistics)

indicates an exercise in which you are instructed to use a spreadsheet.

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56

CHAPTER 1

Functions, Graphs, and Limits

Section 1.3

Lines in the Plane and Slope

■ Use the slope-intercept form of a linear equation to sketch graphs. ■ Find slopes of lines passing through two points. ■ Use the point-slope form to write equations of lines. ■ Find equations of parallel and perpendicular lines. ■ Use linear equations to model and solve real-life problems.

Using Slope

TECHNOLOGY On most graphing utilities, the display screen is twothirds as high as it is wide. On such screens, you can obtain a graph with a true geometric perspective by using a square setting—one in which Ymax  Ymin 2  . X max  X min 3

The simplest mathematical model for relating two variables is the linear equation y  mx  b. The equation is called linear because its graph is a line. (In this text, the term line is used to mean straight line.) By letting x  0, you can see that the line crosses the y-axis at y  b, as shown in Figure 1.31. In other words, the y-intercept is 共0, b兲. The steepness or slope of the line is m. y  mx  b y-intercept

Slope

One such setting is shown below. Notice that the x and y tick marks are equally spaced on a square setting, but not on a standard setting. 4

The slope of a line is the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right, as shown in Figure 1.31. y

y

y = mx + b

y = mx + b y-intercept

m units, m>0

(0, b)

1 unit

(0, b)

m units, m 0



b. Because this function is defined for x < 1 and for x ≥ 1, the domain is the entire set of real numbers. This function is called a piecewise-defined function because it is defined by two or more equations over a specified domain. When x ≥ 1, the function behaves as in part (a). For x < 1, the value of 1 ⫺ x is positive, and therefore the range of the function is y ≥ 0 or 关0, ⬁兲, as shown in Figure 1.45(b). A function is one-to-one if to each value of the dependent variable in the range there corresponds exactly one value of the independent variable. For instance, the function in Example 2(a) is one-to-one, whereas the function in Example 2(b) is not one-to-one. Geometrically, a function is one-to-one if every horizontal line intersects the graph of the function at most once. This geometrical interpretation is the Horizontal Line Test for one-to-one functions. So, a graph that represents a oneto-one function must satisfy both the Vertical Line Test and the Horizontal Line Test.

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72

CHAPTER 1

Functions, Graphs, and Limits

Function Notation When using an equation to define a function, you generally isolate the dependent variable on the left. For instance, writing the equation x ⫹ 2y ⫽ 1 as y⫽

1⫺x 2

indicates that y is the dependent variable. In function notation, this equation has the form f 共x兲 ⫽

1 ⫺ x. 2

Function notation

The independent variable is x, and the name of the function is “f.” The symbol f 共x兲 is read as “f of x,” and it denotes the value of the dependent variable. For instance, the value of f when x ⫽ 3 is f 共3兲 ⫽

1 ⫺ 共3兲 ⫺2 ⫽ ⫽ ⫺1. 2 2

The value f 共3兲 is called a function value, and it lies in the range of f. This means that the point 共3, f 共3兲兲 lies on the graph of f. One of the advantages of function notation is that it allows you to be less wordy. For instance, instead of asking “What is the value of y when x ⫽ 3?” you can ask “What is f 共3兲?”

Example 3

(−1, 7)

f(x)

f(x) =

2x 2

Evaluating a Function

Find the values of the function f 共x兲 ⫽ 2x 2 ⫺ 4x ⫹ 1 when x is ⫺1, 0, and 2. Is f one-to-one?

− 4x + 1

SOLUTION

7

When x ⫽ ⫺1, the value of f is

f 共⫺1兲 ⫽ 2共⫺1兲2 ⫺ 4共⫺1兲 ⫹ 1 ⫽ 2 ⫹ 4 ⫹ 1 ⫽ 7.

6 5

When x ⫽ 0, the value of f is

4

f 共0兲 ⫽ 2共0兲2 ⫺ 4共0兲 ⫹ 1 ⫽ 0 ⫺ 0 ⫹ 1 ⫽ 1. When x ⫽ 2, the value of f is

(2, 1) (0, 1) −1

−1

FIGURE 1.46

2

3

x

f 共2兲 ⫽ 2共2兲2 ⫺ 4共2兲 ⫹ 1 ⫽ 8 ⫺ 8 ⫹ 1 ⫽ 1. Because two different values of x yield the same value of f 共x兲, the function is not one-to-one, as shown in Figure 1.46.

✓CHECKPOINT 3 Find the values of f 共x兲 ⫽ x2 ⫺ 5x ⫹ 1 when x is 0, 1, and 4. Is f one-to-one?



STUDY TIP You can use the Horizontal Line Test to determine whether the function in Example 3 is one-to-one. Because the line y ⫽ 1 intersects the graph of the function twice, the function is not one-to-one.

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SECTION 1.4

73

Functions

Example 3 suggests that the role of the variable x in the equation f 共x兲 ⫽ 2x 2 ⫺ 4x ⫹ 1 is simply that of a placeholder. Informally, f could be defined by the equation f 共䊏兲 ⫽ 2共䊏兲2 ⫺ 4共䊏兲 ⫹ 1. TECHNOLOGY Most graphing utilities can be programmed to evaluate functions. The program depends on the calculator used. The pseudocode below can be translated into a program for a graphing utility. (Appendix E lists the program for several models of graphing utilities.) Program • Label a. • Input x. • Display function value. • Goto a. To use this program, enter a function. Then run the program—it will allow you to evaluate the function at several values of x.

To evaluate f (⫺2兲, simply place ⫺2 in each set of parentheses. f 共⫺2兲 ⫽ 2共⫺2兲2 ⫺ 4共⫺2兲 ⫹ 1 ⫽ 8 ⫹ 8 ⫹ 1 ⫽ 17 The ratio in Example 4(b) is called a difference quotient. In Section 2.1, you will see that it has special significance in calculus.

Example 4

Evaluating a Function

Let f 共x兲 ⫽ x 2 ⫺ 4x ⫹ 7, and find a. f 共x ⫹ ⌬x兲

b.

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 . ⌬x

SOLUTION

a. To evaluate f at x ⫹ ⌬x, substitute x ⫹ ⌬x for x in the original function, as shown. f 共x ⫹ ⌬x兲 ⫽ 共x ⫹ ⌬x兲2 ⫺ 4共x ⫹ ⌬x兲 ⫹ 7 ⫽ x 2 ⫹ 2x ⌬x ⫹ 共⌬x兲2 ⫺ 4x ⫺ 4 ⌬x ⫹ 7 b. Using the result of part (a), you can write f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x 关共x ⫹ ⌬x兲2 ⫺ 4共x ⫹ ⌬x兲 ⫹ 7兴 ⫺ 关x 2 ⫺ 4x ⫹ 7兴 ⫽ ⌬x 2 ⫹ 2x ⌬x ⫹ 共⌬x兲 2 ⫺ 4x ⫺ 4 ⌬x ⫹ 7 ⫺ x 2 ⫹ 4x ⫺ 7 x ⫽ ⌬x 2 2x ⌬x ⫹ 共⌬x兲 ⫺ 4 ⌬x ⫽ ⌬x ⫽ 2x ⫹ ⌬x ⫺ 4, ⌬x ⫽ 0.

✓CHECKPOINT 4 Let f 共x兲 ⫽ x2 ⫺ 2x ⫹ 3, and find (a) f 共x ⫹ ⌬ x兲 and (b)

f 共x ⫹ ⌬ x兲 ⫺ f 共x兲 . ⌬x



Although f is often used as a convenient function name and x as the independent variable, you can use other symbols. For instance, the following equations all define the same function. f 共x兲 ⫽ x 2 ⫺ 4x ⫹ 7 f 共t兲 ⫽ t 2 ⫺ 4t ⫹ 7 g共s兲 ⫽ s 2 ⫺ 4s ⫹ 7

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74

CHAPTER 1

Functions, Graphs, and Limits

Combinations of Functions x

Two functions can be combined in various ways to create new functions. For instance, if f 共x兲 ⫽ 2x ⫺ 3 and g共x兲 ⫽ x 2 ⫹ 1, you can form the following functions.

Input

f 共x兲 ⫹ g共x兲 ⫽ 共2x ⫺ 3兲 ⫹ 共x 2 ⫹ 1兲 ⫽ x 2 ⫹ 2x ⫺ 2 f 共x兲 ⫺ g共x兲 ⫽ 共2x ⫺ 3兲 ⫺ 共x 2 ⫹ 1兲 ⫽ ⫺x 2 ⫹ 2x ⫺ 4 f 共x兲g共x) ⫽ 共2x ⫺ 3兲共x 2 ⫹ 1兲 ⫽ 2x 3 ⫺ 3x 2 ⫹ 2x ⫺ 3 f 共x兲 2x ⫺ 3 ⫽ 2 g共x兲 x ⫹1

Function g

Output g(x)

Sum Difference Product Quotient

You can combine two functions in yet another way called a composition. The resulting function is a composite function. Definition of Composite Function

The function given by 共 f ⬚ g兲共x兲 ⫽ f 共g共x兲兲 is the composite of f with g. The domain of 共 f ⬚ g兲 is the set of all x in the domain of g such that g共x兲 is in the domain of f, as indicated in Figure 1.47.

Input Function f

The composite of f with g may not be equal to the composite of g with f, as shown in the next example. Output

Example 5 f(g(x))

FIGURE 1.47

Forming Composite Functions

Let f 共x兲 ⫽ 2x ⫺ 3 and g共x兲 ⫽ x 2 ⫹ 1, and find a. f 共 g共x兲兲

b. g共 f 共x兲兲.

SOLUTION

a. The composite of f with g is given by f 共 g共x兲兲 ⫽ 2共 g共x兲兲 ⫺ 3 ⫽ 2共x 2 ⫹ 1兲 ⫺ 3 ⫽ 2x 2 ⫺ 1. b. The composite of g with f is given by g共 f 共x兲兲 ⫽ 共 f 共x兲兲 2 ⫹ 1 STUDY TIP The results of f 共g共x兲兲 and g共 f 共x兲兲 are different in Example 5. You can verify this by substituting specific values of x into each function and comparing the results.

Evaluate f at g共x兲. Substitute x 2 ⫹ 1 for g共x兲. Simplify.

Evaluate g at f 共x兲.

⫽ 共2x ⫺ 3兲 ⫹ 1

Substitute 2x ⫺ 3 for f 共x兲.

⫽ 4x 2 ⫺ 12x ⫹ 10.

Simplify.

2

✓CHECKPOINT 5 Let f 共x兲 ⫽ 2x ⫹ 1 and g共x兲 ⫽ x2 ⫹ 2, and find a. f 共g共 x兲兲

b. g 共 f 共 x兲兲.



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SECTION 1.4

Functions

75

Inverse Functions Informally, the inverse function of f is another function g that “undoes” what f has done. f

g

x STUDY TIP Don’t be confused by the use of the superscript ⫺1 to denote the inverse function f ⫺1. In this text, whenever f ⫺1 is written, it always refers to the inverse function of f and not to the reciprocal of f 共x兲.

f 共x兲

Definition of Inverse Function

Let f and g be two functions such that f 共 g共x兲兲 ⫽ x for each x in the domain of g and g共 f 共x兲兲 ⫽ x for each x in the domain of f. Under these conditions, the function g is the inverse function of f. The function g is denoted by f ⫺1, which is read as “ f-inverse.” So, f 共 f ⫺1共x兲兲 ⫽ x

y = f(x)

y

y=x

and

f ⫺1共 f 共x兲兲 ⫽ x.

The domain of f must be equal to the range of f ⫺1, and the range of f must be equal to the domain of f ⫺1.

Example 6 (a, b)

g共 f 共x兲兲 ⫽ x

y = f −1(x)

Finding Inverse Functions

Several functions and their inverse functions are shown below. In each case, note that the inverse function “undoes” the original function. For instance, to undo multiplication by 2, you should divide by 2.

(b, a) x

F I G U R E 1 . 4 8 The graph of f ⫺1 is a reflection of the graph of f in the line y ⫽ x.

STUDY TIP You can verify that the functions in Example 6 are inverse functions by showing that f ( f ⫺ 1(x)) ⫽ x and f ⫺1共 f 共x兲兲 ⫽ x. For Example 6(a), you obtain the following. f 共 f ⫺1共x兲兲 ⫽ f 共12 x兲 ⫽ 2共12x兲 ⫽ x f ⫺1共 f 共x兲兲 ⫽ f ⫺1 共2x兲 ⫽ 12 共2x兲 ⫽ x

a. f 共x兲 ⫽ 2x

f ⫺1共x兲 ⫽ 12 x

b. f 共x兲 ⫽ 13 x

f ⫺1共x兲 ⫽ 3x

c. f 共x兲 ⫽ x ⫹ 4

f ⫺1共x兲 ⫽ x ⫺ 4

d. f 共x兲 ⫽ 2x ⫺ 5

f ⫺1共x兲 ⫽ 12共x ⫹ 5兲

e. f 共x兲 ⫽ x 3

3 x f ⫺1共x兲 ⫽ 冪

f. f 共x兲 ⫽

1 x

f ⫺1共x兲 ⫽

1 x

✓CHECKPOINT 6 Informally find the inverse function of each function. a. f 共x兲 ⫽ 15 x

b. f 共x兲 ⫽ 3x ⫹ 2



The graphs of f and f ⫺1 are mirror images of each other (with respect to the line y ⫽ x兲, as shown in Figure 1.48. Try using a graphing utility to confirm this for each of the functions given in Example 6.

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76

CHAPTER 1

Functions, Graphs, and Limits

The functions in Example 6 are simple enough so that their inverse functions can be found by inspection. The next example demonstrates a strategy for finding the inverse functions of more complicated functions.

Example 7

Finding an Inverse Function

Find the inverse function of f 共x兲 ⫽ 冪2x ⫺ 3. SOLUTION

for y.

f 共x兲 ⫽ 冪2x ⫺ 3 y ⫽ 冪2x ⫺ 3 x ⫽ 冪2y ⫺ 3 x 2 ⫽ 2y ⫺ 3 x 2 ⫹ 3 ⫽ 2y x2 ⫹ 3 ⫽y 2

2 f −1(x) = x + 3 2

y 6

4

Begin by replacing f 共x兲 with y. Then, interchange x and y and solve

f ⫺1共䊏兲 ⫽

(1, 2) 2

f(x) =

( 0(

4

Interchange x and y. Square each side. Add 3 to each side. Divide each side by 2.

6

FIGURE 1.49

共䊏兲2 ⫹ 3 . 2

Using x as the independent variable, you can write

2x − 3

(2, 1) 3 2,

Replace f 共x兲 with y.

So, the inverse function has the form

y=x

(0, 32 (

Write original function.

x

f ⫺1共x兲 ⫽

x2 ⫹ 3 , 2

x ≥ 0.

In Figure 1.49, note that the domain of f ⫺1 coincides with the range of f.

✓CHECKPOINT 7 Find the inverse function of f 共x兲 ⫽ x2 ⫹ 2 for x ≥ 0. TECHNOLOGY A graphing utility can help you check that the graphs of f and f ⫺1 are reflections of each other in the line y ⫽ x. To do this, graph y ⫽ f 共x兲, y ⫽ f ⫺1共x兲, and y ⫽ x in the same viewing window, using a square setting.



After you have found an inverse function, you should check your results. You can check your results graphically by observing that the graphs of f and f ⫺1 are reflections of each other in the line y ⫽ x. You can check your results algebraically by evaluating f 共 f ⫺1共x兲兲 and f ⫺1共 f 共x兲兲—both should be equal to x. Check that f 共 f ⫺1共x兲兲 ⫽ x f 共 f ⫺1共x兲兲 ⫽ f ⫽

冢x

2

⫹3 2

Check that f ⫺1共 f 共x兲兲 ⫽ x



冪冢

f ⫺1共 f 共x兲兲 ⫽ f ⫺1共冪2x ⫺ 3 兲

x2 ⫹ 3 2 ⫺3 2

⫽ 冪x 2 ⫽ x, x ≥ 0





共冪2x ⫺ 3 兲2 ⫹ 3 2

2x ⫽ 2 ⫽ x,

x ≥

3 2

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SECTION 1.4

Functions

77

Not every function has an inverse function. In fact, for a function to have an inverse function, it must be one-to-one.

Example 8

A Function That Has No Inverse Function

Show that the function f 共x兲 ⫽ x 2 ⫺ 1 has no inverse function. (Assume that the domain of f is the set of all real numbers.) y

SOLUTION

(−2, 3)

Begin by sketching the graph of f, as shown in Figure 1.50. Note that

f 共2兲 ⫽ 共2兲2 ⫺ 1 ⫽ 3

(2, 3)

and 2

y=3

f 共⫺2兲 ⫽ 共⫺2兲2 ⫺ 1 ⫽ 3.

1

−2

−1

1

f (x) =

2

x2

x

−1

F I G U R E 1 . 5 0 f is not one-to-one and has no inverse function.

So, f does not pass the Horizontal Line Test, which implies that it is not one-toone, and therefore has no inverse function. The same conclusion can be obtained by trying to find the inverse function of f algebraically. f 共x兲 ⫽ x 2 ⫺ 1 y ⫽ x2 ⫺ 1 x ⫽ y2 ⫺ 1 x ⫹ 1 ⫽ y2 ± 冪x ⫹ 1 ⫽ y

Write original function. Replace f 共x兲 with y. Interchange x and y. Add 1 to each side. Take square root of each side.

The last equation does not define y as a function of x, and so f has no inverse function.

✓CHECKPOINT 8 Show that the function f 共x兲 ⫽ x2 ⫹ 4 has no inverse function.



CONCEPT CHECK 1. Explain the difference between a relation and a function. 2. In your own words, explain the meanings of domain and range. 3. Is every relation a function? Explain. 4. Describe how to find the inverse of a function given by an equation in x and y.

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78

CHAPTER 1

Skills Review 1.4

Functions, Graphs, and Limits The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 0.3 and 0.5.

In Exercises 1–6, simplify the expression. 1. 5共⫺1兲2 ⫺ 6共⫺1兲 ⫹ 9

2. 共⫺2兲3 ⫹ 7共⫺2兲2 ⫺ 10

4. 共3 ⫺ x兲 ⫹ 共x ⫹ 3兲3

5.

3. 共x ⫺ 2兲2 ⫹ 5x ⫺ 10

1 1 ⫺ 共1 ⫺ x兲

6. 1 ⫹

x⫺1 x

In Exercises 7–12, solve for y in terms of x. 7. 2x ⫹ y ⫺ 6 ⫽ 11

8. 5y ⫺ 6x 2 ⫺ 1 ⫽ 0

10. y 2 ⫺ 4x 2 ⫽ 2

11. x ⫽

2y ⫺ 1 4

Exercises 1.4

1. x 2 ⫹ y 2 ⫽ 4 3.



8. x 2y ⫺ x 2 ⫹ 4y ⫽ 0

9. f 共x兲 ⫽ 2x 2 ⫺ 5x ⫹ 1

3

2

2

1

12. f 共x兲 ⫽ 冪9 ⫺ x 2

x



x 13. f 共x兲 ⫽ 冪x ⫺ 4

x < 0 x ≥ 0

3x ⫹ 2, 14. f 共x兲 ⫽ 2 ⫺ x,

x⫺2 x⫹4

16. f 共x兲 ⫽

x2 1⫺x

x 1

(a) f 共0兲

4

x

(b) f 共x ⫺ 1兲

(c) f 共x ⫹ ⌬x兲

22. f 共x兲 ⫽ x ⫺ 4x ⫹ 1 2

1 (a) g共4 兲

(b) f 共c ⫹ 2兲

(c) f 共x ⫹ ⌬x兲

(b) g共x ⫹ 4兲

(c) g共x ⫹ ⌬x兲 ⫺ g共x兲

ⱍⱍ

24. f 共x兲 ⫽ x ⫹ 4 (a) f 共⫺2兲

y

1

x

3

21. f 共x兲 ⫽ 3x ⫺ 2

(a) f 共⫺1兲

18. f 共x兲 ⫽ 冪2x ⫺ 3 y

1

2

23. g共x兲 ⫽ 1兾x

In Exercises 17–20, find the domain and range of the function. Use interval notation to write your result. 17. f 共x兲 ⫽ x 3

−1

1

In Exercises 21–24, evaluate the function at the specified values of the independent variable. Simplify the result.

10. f 共x兲 ⫽ 5x 3 ⫹ 6x 2 ⫺ 1

ⱍxⱍ

−1



1

In Exercises 9–16, use a graphing utility to graph the function. Then determine the domain and range of the function.

15. f 共x兲 ⫽

y 3

6. x2 ⫹ y2 ⫹ 2x ⫽ 0

7. y ⫽ x ⫹ 2



20. f 共x兲 ⫽ x ⫺ 2

y

4. 3x ⫺ 2y ⫹ 5 ⫽ 0

5. x 2 ⫹ y ⫽ 4

11. f 共x兲 ⫽

19. f 共x兲 ⫽ 4 ⫺ x 2

2. x ⫹ y 2 ⫽ 4

⫺ 6y ⫽ ⫺3



3 2y ⫺ 1 12. x ⫽ 冪

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 8, decide whether the equation defines y as a function of x. 1 2x

9. 共 y ⫺ 3兲2 ⫽ 5 ⫹ 共x ⫹ 1兲2

(b) f 共x ⫹ 2兲

(c) f 共x ⫹ ⌬x兲 ⫺ f 共x兲

3

In Exercises 25–30, evaluate the difference quotient and simplify the result.

2

25. f 共x兲 ⫽ x2 ⫺ 5x ⫹ 2 f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

1

−1 1

2

3

26. h 共x兲 ⫽ x 2 ⫹ x ⫹ 3 h 共2 ⫹ ⌬x兲 ⫺ h 共2兲 ⌬x

x

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.4 27. g共x兲 ⫽ 冪x ⫹ 1

28. f 共x兲 ⫽

In Exercises 41– 44, show that f and g are inverse functions by showing that f 冇 g 冇x冈冈 ⴝ x and g 冇f 冇x冈冈 ⴝ x. Then sketch the graphs of f and g on the same coordinate axes.

冪x

f 共x兲 ⫺ f 共2兲 x⫺2

g 共x ⫹ ⌬x兲 ⫺ g 共x兲 ⌬x 29. f 共x兲 ⫽

1

1 x⫺2

30. f 共x兲 ⫽

1 x⫹4

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

79

Functions

41. f 共x兲 ⫽ 5x ⫹ 1,

g共x兲 ⫽

x⫺1 5

1 42. f 共x兲 ⫽ , x

g共x兲 ⫽

1 x

43. f 共x兲 ⫽ 9 ⫺ x 2,

g共x兲 ⫽ 冪9 ⫺ x,

x ≥ 0,

x ≤ 9

In Exercises 31–34, use the Vertical Line Test to determine whether y is a function of x.

44. f 共x兲 ⫽ 1 ⫺ x 3,

31. x 2 ⫹ y 2 ⫽ 9

In Exercises 45–52, find the inverse function of f. Then use a graphing utility to graph f and f ⫺1 on the same coordinate axes.

32. x ⫺ xy ⫹ y ⫹ 1 ⫽ 0

y

y

2 1 −2 −1

1

3

45. f 共x兲 ⫽ 2x ⫺ 3

46. f 共x兲 ⫽ 7 ⫺ x

2

47. f 共x兲 ⫽ x

48. f 共x兲 ⫽ x 3

x

2

−2

−1

33. x 2 ⫽ xy ⫺ 1

2

−1

y

3

3

ⱍⱍ

1 1

2

3

x

1

2

x

−1

In Exercises 35–38, find (a) f 冇x冈 1 g冇x冈, (b) f 冇x冈 ⭈ g冇x冈, (c) f 冇x冈/g冇x冈, (d) f 冇 g 冇x冈冈, and (e) g 冇f 冇x冈冈, if defined. 35. f 共x兲 ⫽ 2x ⫺ 5 g共x兲 ⫽ 5 37. f 共x兲 ⫽ x 2 ⫹ 1 g共x兲 ⫽ x ⫺ 1

36. f 共x兲 ⫽ x 2 ⫹ 5 g共x兲 ⫽ 冪1 ⫺ x x x⫹1 g共x兲 ⫽ x 3

38. f 共x兲 ⫽

39. Given f 共x兲 ⫽ 冪x and g共x兲 ⫽ x 2 ⫺ 1, find the composite functions. (a) f 共 g共1兲兲

(b) g共 f 共1兲兲

(c) g共 f 共0兲兲

(d) f 共 g共⫺4兲兲

(e) f 共 g共x兲兲

(f) g共 f 共x兲兲

40. Given f 共x兲 ⫽ 1兾x and g共x兲 ⫽ x 2 ⫺ 1, find the composite functions. (a) f 共 g共2兲兲

(b) g共 f 共2兲兲

(e) f 共 g共x兲兲

(f) g共 f 共x兲兲

(c) f 共 g共1兾冪2 兲兲

5

49. f 共x兲 ⫽ 冪9 ⫺ x 2,

0 ≤ x ≤ 3

50. f 共x兲 ⫽

冪x 2

x ≥ 2

51. f 共x兲 ⫽

x 2兾3,

⫺ 4,

52. f 共x兲 ⫽ x 3兾5

x ≥ 0

In Exercises 53– 58, use a graphing utility to graph the function. Then use the Horizontal Line Test to determine whether the function is one-to-one. If it is, find its inverse function.

1

2

−3 − 2 − 1

x

34. x ⫽ y

y

3 1⫺x g共x兲 ⫽ 冪

(d) g共 f 共1兾冪2 兲兲

53. f 共x兲 ⫽ 3 ⫺ 7x

54. f 共x兲 ⫽ 冪x ⫺ 2

55. f 共x兲 ⫽ x

56. f 共x兲 ⫽ x 4

2





57. f 共x兲 ⫽ x ⫹ 3

58. f 共x兲 ⫽ ⫺5

59. Use the graph of f 共x兲 ⫽ 冪x below to sketch the graph of each function. y

(a) y ⫽ 冪x ⫹ 2 (b) y ⫽ ⫺ 冪x

3

(c) y ⫽ 冪x ⫺ 2

2

(d) y ⫽ 冪x ⫹ 3

1

f(x) =

x

(e) y ⫽ 冪x ⫺ 4 1

(f) y ⫽ 2冪x

2

3

x

4

ⱍⱍ

60. Use the graph of f 共x兲 ⫽ x below to sketch the graph of each function.

ⱍⱍ

y

(a) y ⫽ x ⫹ 3 (b) y ⫽

⫺ 12

ⱍxⱍ

3

ⱍ ⱍ (d) y ⫽ ⱍx ⫹ 1ⱍ ⫺ 1 (e) y ⫽ 2ⱍxⱍ (c) y ⫽ x ⫺ 2

f(x) = ⏐x⏐

2 1 −2

−1

1

2

−1

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

x

80

CHAPTER 1

Functions, Graphs, and Limits

61. Use the graph of f 共x兲 ⫽ x 2 to write an equation for each function whose graph is shown. (a)

(b)

y 9

−9

y

−6

x

−3

(−6, − 3)

−3

−6

−3

3

where t ⫽ 1 represents 2001. During the same seven-year period, the sales R2 (in thousands of dollars) for the second restaurant can be modeled by

−9

(−3, 0)

R2 ⫽ 458 ⫹ 0.78t, t ⫽ 1, 2, 3, 4, 5, 6, 7.

62. Use the graph of f 共x兲 ⫽ to write an equation for each function whose graph is shown. x3

(a)

(b)

y

y

3 2

2

(2, 1) 1

−1

1

2

3

x

−3 −2 −1

−2

−2

−3

−3

1

2

3

x

(1, − 2)

63. Prescription Drugs The amounts d (in billions of dollars) spent on prescription drugs in the United States from 1991 through 2005 (see figure) can be approximated by the model d共t兲 ⫽

⫺ 0.3t ⫹ 45, 冦yy ⫽⫽ 0.68t 16.7t ⫺ 45, 2

1 ≤ t ≤ 8 9 ≤ t ≤ 15

(a) Express the total cost C as a function of x, the number of games sold. (b) Find a formula for the average cost per unit C ⫽ C兾x. (c) The selling price for each game is $4.95. How many units must be sold before the average cost per unit falls below the selling price? 67. Demand The demand function for a commodity is p⫽

14.75 , x ≥ 0 1 ⫹ 0.01x

where p is the price per unit and x is the number of units sold.

where t represents the year, with t ⫽ 1 corresponding to 1991. (Source: U.S. Centers for Medicare & Medicaid Services) d 210

Amount spent (in billions of dollars)

Write a function that represents the total sales for the two restaurants. Use a graphing utility to graph the total sales function. 66. Cost The inventor of a new game believes that the variable cost for producing the game is $1.95 per unit. The fixed cost is $6000.

3

1 −3

65. Owning a Business You own two restaurants. From 2001 through 2007, the sales R1 (in thousands of dollars) for one restaurant can be modeled by R1 ⫽ 690 ⫺ 8t ⫺ 0.8t2, t ⫽ 1, 2, 3, 4, 5, 6, 7

−6 x

64. Real Estate Express the value V of a real estate firm in terms of x, the number of acres of property owned. Each acre is valued at $2500 and other company assets total $750,000.

180 150

(a) Find x as a function of p. (b) Find the number of units sold when the price is $10. 68. Cost A power station is on one side of a river that is 12 mile wide. A factory is 3 miles downstream on the other side of the river (see figure). It costs $15/ft to run the power lines on land and $20/ft to run them under water. Express the cost C of running the lines from the power station to the factory as a function of x.

120 90 60 30

x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

t

Year (1 ↔ 1991)

Factory

3−x

1 2

Power station

(a) Use a graphing utility to graph the function. (b) Find the amounts spent on prescription drugs in 1997, 2000, and 2004.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.4 69. Cost The weekly cost of producing x units in a manufacturing process is given by the function C共x兲 ⫽ 70x ⫹ 375. The number of units produced in t hours is given by x共t兲 ⫽ 40t. Find and interpret C共x共t兲兲. 70. Market Equilibrium The supply function for a product relates the number of units x that producers are willing to supply for a given price per unit p. The supply and demand functions for a market are p⫽

2 x⫹4 5

p⫽⫺

16 x ⫹ 30. 25

Supply Demand

(a) Use a graphing utility to graph the supply and demand functions in the same viewing window. (b) Use the trace feature of the graphing utility to find the equilibrium point for the market. (c) For what values of x does the demand exceed the supply? (d) For what values of x does the supply exceed the demand? 71. Profit A manufacturer charges $90 per unit for units that cost $60 to produce. To encourage large orders from distributors, the manufacturer will reduce the price by $0.01 per unit for each unit in excess of 100 units. (For example, an order of 101 units would have a price of $89.99 per unit, and an order of 102 units would have a price of $89.98 per unit.) This price reduction is discontinued when the price per unit drops to $75. (a) Express the price per unit p as a function of the order size x. (b) Express the profit P as a function of the order size x.

(b) Complete the table. n

100

125

150

F共t兲 ⫽ 98 ⫹

(a) Express the revenue R for the bus company as a function of n.

250

3 t⫹1

where F is the temperature in degrees Fahrenheit and t is the time in hours since the drug was administered. Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. For what values of t do you think this function would be valid? Explain. In Exercises 75–80, use a graphing utility to graph the function. Then use the zoom and trace features to find the zeros of the function. Is the function one-to-one? 75. f 共x兲 ⫽ 9x ⫺ 4x 2



76. f 共x兲 ⫽ 2 3x 2 ⫺ 77. g共t兲 ⫽

6 x



t⫹3 1⫺t

78. h共x兲 ⫽ 6x 3 ⫺ 12x 2 ⫹ 4 79. g共x兲 ⫽ x 2冪x 2 ⫺ 4 80. g共x兲 ⫽

ⱍ ⱍ 1 2 x ⫺4 2

Business Capsule CardSenders is a homebased greeting card service for businesses. Cap Poore bought the company in 2003, which has licensees operating in the United Kingdom, Canada, Asia, and Mexico. Start-up costs are $6,900.00 for licensees.

(c) Write the profit P as a function of x.

where n is the number of people.

225

74. Medicine The temperature of a patient after being given a fever-reducing drug is given by

(b) Write the revenue R as a function of x.

n ≥ 80

200

(c) Criticize the formula for the rate. Would you use this formula? Explain your reasoning.

(a) Write the total cost C as a function of x.

r ⫽ 15 ⫺ 0.05共n ⫺ 80兲,

175

R

72. Cost, Revenue, and Profit A company invests $98,000 for equipment to produce a new product. Each unit of the product costs $12.30 and is sold for $17.98. Let x be the number of units produced and sold.

73. MAKE A DECISION: REVENUE For groups of 80 or more people, a charter bus company determines the rate r (in dollars per person) according to the formula

81

Functions

Photo courtesy of Cap Poore

81. Research Project Use your school’s library, the Internet, or some other reference source to find information about the start-up costs of beginning a business, such as the example above. Write a short paper about the company.

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82

CHAPTER 1

Functions, Graphs, and Limits

Section 1.5 ■ Find limits of functions graphically and numerically.

Limits

■ Use the properties of limits to evaluate limits of functions. ■ Use different analytic techniques to evaluate limits of functions. ■ Evaluate one-sided limits. ■ Recognize unbounded behavior of functions.

The Limit of a Function

w=0 s

w=3

w = 7.5

w = 9.5

In everyday language, people refer to a speed limit, a wrestler’s weight limit, the limit of one’s endurance, or stretching a spring to its limit. These phrases all suggest that a limit is a bound, which on some occasions may not be reached but on other occasions may be reached or exceeded. Consider a spring that will break only if a weight of 10 pounds or more is attached. To determine how far the spring will stretch without breaking, you could attach increasingly heavier weights and measure the spring length s for each weight w, as shown in Figure 1.51. If the spring length approaches a value of L, then it is said that “the limit of s as w approaches 10 is L.” A mathematical limit is much like the limit of a spring. The notation for a limit is lim f 共x兲 ⫽ L

w = 9.999

F I G U R E 1 . 5 1 What is the limit of s as w approaches 10 pounds?

x→c

which is read as “the limit of f 共x兲 as x approaches c is L.”

Example 1

Finding a Limit

Find the limit: lim 共x 2 ⫹ 1兲. x→1

y

lim (x 2 + 1) = 2

x→1

Let f 共x兲 ⫽ x 2 ⫹ 1. From the graph of f in Figure 1.52, it appears that f 共x兲 approaches 2 as x approaches 1 from either side, and you can write

SOLUTION

lim 共x 2 ⫹ 1兲 ⫽ 2.

4

x→1

The table yields the same conclusion. Notice that as x gets closer and closer to 1, f 共x兲 gets closer and closer to 2.

3

−2

−1

FIGURE 1.52

1

x approaches 1.

x approaches 1.

(1, 2)

2

2

x

x

0.900

0.990

0.999

1.000

1.001

1.010

1.100

f 共x兲

1.810

1.980

1.998

2.000

2.002

2.020

2.210

f 共x兲 approaches 2.

f 共x兲 approaches 2.

✓CHECKPOINT 1 Find the limit: lim 共2x ⫹ 4兲. x→1



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.5

Example 2 y

83

Limits

Finding Limits Graphically and Numerically

Find the limit: lim f 共x兲. x→1

3

2 lim x − 1 = 2 x→1 x − 1

2

(1, 2)

b. f 共x兲 ⫽

2

3

x

⏐x − 1⏐ lim does not exist. x→1 x − 1 (1, 1)

1

1

c. f 共x兲 ⫽

x⫺1

2

x

0.900

0.990

0.999

1.000

1.001

1.010

1.100

f 共x兲

1.900

1.990

1.999

?

2.001

2.010

2.100

f 共x兲 approaches 2.

x approaches 1.

x approaches 1.

y

x

lim f (x) = 1 x→1

f 共x兲

0.900

0.990

0.999

1.000

1.001

1.010

1.100

⫺1.000

⫺1.000

⫺1.000

?

1.000

1.000

1.000

f 共x兲 approaches ⫺1.

(1, 1)

1

2

3

(c)

x

x approaches 1.

✓CHECKPOINT 2 Find the limit: lim f 共x兲. x→2

x2 ⫺ 4 x⫺2 x⫺2 b. f 共x兲 ⫽ x⫺2 c. f 共x兲 ⫽

x approaches 1.

x

0.900

0.990

0.999

1.000

1.001

1.010

1.100

f 共x兲

0.900

0.990

0.999

?

1.001

1.010

1.100

f 共x兲 approaches 1.

a. f 共x兲 ⫽

f 共x兲 approaches 1.

c. From the graph of f, in Figure 1.53(c), it appears that f 共x兲 approaches 1 as x approaches 1 from either side. This conclusion is reinforced by the table. It does not matter that f 共1兲 ⫽ 0. The limit depends only on values of f 共x兲 near 1, not at 1.

FIGURE 1.53



x⫽1 x⫽1

b. From the graph of f, in Figure 1.53(b), you can see that f 共x兲 ⫽ ⫺1 for all values to the left of x ⫽ 1 and f 共x兲 ⫽ 1 for all values to the right of x ⫽ 1. So, f 共x兲 is approaching a different value from the left of x ⫽ 1 than it is from the right of x ⫽ 1. In such situations, we say that the limit does not exist. This conclusion is reinforced by the table.

(b)

1

冦x,0,

x approaches 1.

x approaches 1.

f 共x兲 approaches 2.

x

(1, − 1)

2

ⱍx ⫺ 1ⱍ

a. From the graph of f, in Figure 1.53(a), it appears that f 共x兲 approaches 2 as x approaches 1 from either side. A missing point is denoted by the open dot on the graph. This conclusion is reinforced by the table. Be sure you see that it does not matter that f 共x兲 is undefined when x ⫽ 1. The limit depends only on values of f 共x兲 near 1, not at 1.

(a)

y

x ⫺1 x⫺1

SOLUTION

1

1

a. f 共x兲 ⫽

2

f 共x兲 approaches 1.



冦0,x , 2

x⫽2 x⫽2



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

84

CHAPTER 1

Functions, Graphs, and Limits

TECHNOLOGY Try using a graphing utility to determine the following limit. x 3 ⫹ 4x ⫺ 5 x→1 x⫺1 lim

You can do this by graphing f 共x兲 ⫽

x 3 ⫹ 4x ⫺ 5 x⫺1

and zooming in near x ⫽ 1. From the graph, what does the limit appear to be?

There are three important ideas to learn from Examples 1 and 2. 1. Saying that the limit of f 共x兲 approaches L as x approaches c means that the value of f 共x兲 may be made arbitrarily close to the number L by choosing x closer and closer to c. 2. For a limit to exist, you must allow x to approach c from either side of c. If f 共x兲 approaches a different number as x approaches c from the left than it does as x approaches c from the right, then the limit does not exist. [See Example 2(b).] 3. The value of f 共x兲 when x ⫽ c has no bearing on the existence or nonexistence of the limit of f 共x兲 as x approaches c. For instance, in Example 2(a), the limit of f 共x兲 exists as x approaches 1 even though the function f is not defined at x ⫽ 1. Definition of the Limit of a Function

If f 共x兲 becomes arbitrarily close to a single number L as x approaches c from either side, then lim f 共x兲 ⫽ L

x→c

which is read as “the limit of f 共x兲 as x approaches c is L.”

Properties of Limits Many times the limit of f 共x兲 as x approaches c is simply f 共c兲, as shown in Example 1. Whenever the limit of f 共x兲 as x approaches c is lim f 共x兲 ⫽ f 共c兲

x→c

Substitute c for x.

the limit can be evaluated by direct substitution. (In the next section, you will learn that a function that has this property is continuous at c.) It is important that you learn to recognize the types of functions that have this property. Some basic ones are given in the following list. Properties of Limits

Let b and c be real numbers, and let n be a positive integer. 1. lim b ⫽ b x→c

2. lim x ⫽ c x→c

3. lim x n ⫽ c n x→c

n x ⫽冪 n c 4. lim 冪 x→c

In Property 4, if n is even, then c must be positive.

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SECTION 1.5

Limits

85

By combining the properties of limits with the rules for operating with limits shown below, you can find limits for a wide variety of algebraic functions. TECHNOLOGY Symbolic computer algebra systems are capable of evaluating limits. Try using a computer algebra system to evaluate the limit given in Example 3.

Operations with Limits

Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits. lim f 共x兲 ⫽ L and lim g 共x兲 ⫽ K

x→c

x→c

1. Scalar multiple: lim 关bf 共x兲兴 ⫽ bL x→c

2. Sum or difference: lim 关 f 共x兲 ± g共x兲兴 ⫽ L ± K x→c

3. Product: lim 关 f 共x兲 x→c

⭈ g共x兲兴 ⫽ LK

f 共x兲 L ⫽ , provided K ⫽ 0 x→c g共x兲 K

4. Quotient: lim

5. Power: lim 关 f 共x兲兴 n ⫽ Ln x→c

n f 共x兲 ⫽ 冪 n L 6. Radical: lim 冪 x→c

In Property 6, if n is even, then L must be positive. D I S C O V E RY Use a graphing utility to graph y1 ⫽ 1兾x 2. Does y1 approach a limit as x approaches 0? Evaluate y1 ⫽ 1兾x 2 at several positive and negative values of x near 0 to confirm your answer. Does lim 1兾x 2 exist?

Example 3

Finding the Limit of a Polynomial Function

Find the limit: lim 共x 2 ⫹ 2x ⫺ 3兲. x→2

x2

lim 共

x→2

⫹ 2x ⫺ 3兲 ⫽ lim x2 ⫹ lim 2x ⫺ lim 3 x→2

x→2

⫽ 2 2 ⫹ 2共2兲 ⫺ 3 ⫽4⫹4⫺3 ⫽5

x→1

x→2

Apply Property 2. Use direct substitution. Simplify.

✓CHECKPOINT 3 Find the limit: lim 共2x2 ⫺ x ⫹ 4兲. x→1



Example 3 is an illustration of the following important result, which states that the limit of a polynomial function can be evaluated by direct substitution. The Limit of a Polynomial Function

If p is a polynomial function and c is any real number, then lim p共x兲 ⫽ p共c兲.

x→c

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86

CHAPTER 1

Functions, Graphs, and Limits

Techniques for Evaluating Limits Many techniques for evaluating limits are based on the following important theorem. Basically, the theorem states that if two functions agree at all but a single point c, then they have identical limit behavior at x ⫽ c. The Replacement Theorem

Let c be a real number and let f 共x兲 ⫽ g共x兲 for all x ⫽ c. If the limit of g共x兲 exists as x → c, then the limit of f 共x兲 also exists and lim f 共x兲 ⫽ lim g共x兲.

x→c

x→c

To apply the Replacement Theorem, you can use a result from algebra which states that for a polynomial function p, p共c兲 ⫽ 0 if and only if 共x ⫺ c兲 is a factor of p共x兲. This concept is demonstrated in Example 4.

y

3

Example 4 2

Finding the Limit of a Function x3 ⫺ 1 . x→1 x ⫺ 1

Find the limit: lim 1

−2

f(x) =

−1

1

x3 − 1 x−1 x

x 3 ⫺ 1 共x ⫺ 1兲共x 2 ⫹ x ⫹ 1兲 ⫽ x⫺1 x⫺1 共x ⫺ 1兲共x2 ⫹ x ⫹ 1兲 ⫽ x⫺1 ⫽ x2 ⫹ x ⫹ 1, x ⫽ 1

g(x) = x 2 + x + 1 y

2

FIGURE 1.54

Divide out like factor. Simplify.

x3 ⫺ 1 ⫽ lim 共x2 ⫹ x ⫹ 1兲 ⫽ 12 ⫹ 1 ⫹ 1 ⫽ 3 x→1 x ⫺ 1 x→1 lim

1

−1

Factor numerator.

So, the rational function 共x 3 ⫺ 1兲兾共x ⫺ 1兲 and the polynomial function x 2 ⫹ x ⫹ 1 agree for all values of x other than x ⫽ 1, and you can apply the Replacement Theorem.

3

−2

Note that the numerator and denominator are zero when x ⫽ 1. This implies that x ⫺ 1 is a factor of both, and you can divide out this like factor. SOLUTION

1

x

Figure 1.54 illustrates this result graphically. Note that the two graphs are identical except that the graph of g contains the point 共1, 3兲, whereas this point is missing on the graph of f. (In the graph of f in Figure 1.54, the missing point is denoted by an open dot.)

✓CHECKPOINT 4 Find the limit:

x3 ⫺ 8 . x→2 x ⫺ 2 lim



The technique used to evaluate the limit in Example 4 is called the dividing out technique. This technique is further demonstrated in the next example.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.5

Example 5

D I S C O V E RY Use a graphing utility to graph

Using the Dividing Out Technique x2 ⫹ x ⫺ 6 . x→⫺3 x⫹3

Direct substitution fails because both the numerator and the denominator are zero when x ⫽ ⫺3. SOLUTION

Is the graph a line? Why or why not?

x →⫺3

lim 共x ⫹ 3兲 ⫽ 0

x →⫺3

However, because the limits of both the numerator and denominator are zero, you know that they have a common factor of x ⫹ 3. So, for all x ⫽ ⫺3, you can divide out this factor to obtain the following.

1 x

−1

1

2

3

−2

2 f(x) = x + x − 6 x+3

−3

共x ⫺ 2兲共x ⫹ 3兲 x2 ⫹ x ⫺ 6 ⫽ lim x→⫺3 x→⫺3 x⫹3 x⫹3 共x ⫺ 2兲共x ⫹ 3兲 ⫽ lim x→⫺3 x⫹3 ⫽ lim 共x ⫺ 2兲 lim

−1

Factor numerator. Divide out like factor. Simplify.

x→⫺3

−4

(−3, − 5)

lim 共x 2 ⫹ x ⫺ 6兲 ⫽ 0

x2 ⫹ x ⫺ 6 x→⫺3 x⫹3 lim

y

−2

⫽ ⫺5

−5

FIGURE 1.55 when x ⫽ ⫺3.

87

Find the limit: lim

x2 ⫹ x ⫺ 6 . x⫹3

y⫽

Limits

Direct substitution

This result is shown graphically in Figure 1.55. Note that the graph of f coincides with the graph of g共x兲 ⫽ x ⫺ 2, except that the graph of f has a hole at 共⫺3, ⫺5兲.

f is undefined

✓CHECKPOINT 5 x2 ⫹ x ⫺ 12 . x→3 x⫺3

Find the limit: lim

Example 6

STUDY TIP When you try to evaluate a limit and both the numerator and denominator are zero, remember that you must rewrite the fraction so that the new denominator does not have 0 as its limit. One way to do this is to divide out like factors, as shown in Example 5. Another technique is to rationalize the numerator, as shown in Example 6.

x→0

Finding a Limit of a Function

Find the limit: lim

冪x ⫹ 1 ⫺ 1

x

x→0

.

SOLUTION Direct substitution fails because both the numerator and the denominator are zero when x ⫽ 0. In this case, you can rewrite the fraction by rationalizing the numerator.

冪x ⫹ 1 ⫺ 1

x

✓CHECKPOINT 6 Find the limit: lim





冪x ⫹ 1 ⫺ 1

冣冢



冪x ⫹ 1 ⫹ 1 x 冪x ⫹ 1 ⫹ 1 共x ⫹ 1兲 ⫺ 1 ⫽ x共冪x ⫹ 1 ⫹ 1兲 x 1 ⫽ ⫽ , x⫽0 x共冪x ⫹ 1 ⫹ 1兲 冪x ⫹ 1 ⫹ 1



Now, using the Replacement Theorem, you can evaluate the limit as shown.

冪x ⫹ 4 ⫺ 2

x

.

lim



x→0

冪x ⫹ 1 ⫺ 1

x

⫽ lim

x→0

1 冪x ⫹ 1 ⫹ 1



1 1 ⫽ 1⫹1 2

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88

CHAPTER 1

Functions, Graphs, and Limits

One-Sided Limits In Example 2(b), you saw that one way in which a limit can fail to exist is when a function approaches a different value from the left of c than it approaches from the right of c. This type of behavior can be described more concisely with the concept of a one-sided limit. lim f 共x兲 ⫽ L

Limit from the left

lim f 共x兲 ⫽ L

Limit from the right

x→c⫺ x→c⫹

The first of these two limits is read as “the limit of f 共x兲 as x approaches c from the left is L.” The second is read as “the limit of f 共x兲 as x approaches c from the right is L.”

Example 7 y

f(x) =

Finding One-Sided Limits

Find the limit as x → 0 from the left and the limit as x → 0 from the right for the function

⏐2x⏐ x

2

f 共x兲 ⫽

1

ⱍ2xⱍ. x

From the graph of f, shown in Figure 1.56, you can see that f 共x兲 ⫽ ⫺2 for all x < 0. So, the limit from the left is

SOLUTION −2

−1

1

2

−1

x

lim

x→0⫺

ⱍ2xⱍ ⫽ ⫺2. x

Limit from the left

Because f 共x兲 ⫽ 2 for all x > 0, the limit from the right is FIGURE 1.56

TECHNOLOGY On most graphing utilities, the absolute value function is denoted by abs. You can verify the result in Example 7 by graphing y⫽

abs共2x兲 x

in the viewing window ⫺3 ≤ x ≤ 3 and ⫺3 ≤ y ≤ 3.

lim⫹

x→0

ⱍ2xⱍ ⫽ 2.

Limit from the right

x

✓CHECKPOINT 7 Find each limit.

(a) lim⫺ x→2

ⱍx ⫺ 2ⱍ x⫺2

(b) lim⫹ x→2

ⱍx ⫺ 2ⱍ x⫺2



In Example 7, note that the function approaches different limits from the left and from the right. In such cases, the limit of f 共x兲 as x → c does not exist. For the limit of a function to exist as x → c, both one-sided limits must exist and must be equal. Existence of a Limit

If f is a function and c and L are real numbers, then lim f 共x兲 ⫽ L

x→c

if and only if both the left and right limits are equal to L.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.5 y

4

Example 8

f(x) = 4 − x (x < 1)

89

Finding One-Sided Limits

Find the limit of f 共x兲 as x approaches 1.

f(x) = 4x − x 2 (x > 1)

3

Limits

f 共x兲 ⫽

2

冦44x⫺⫺x,x , 2

x < 1 x > 1

Remember that you are concerned about the value of f near x ⫽ 1 rather than at x ⫽ 1. So, for x < 1, f 共x兲 is given by 4 ⫺ x, and you can use direct substitution to obtain

SOLUTION 1

1

2

3

x

5

lim f 共x兲 ⫽ lim⫺ 共4 ⫺ x兲

x→1⫺

lim f(x) = 3

x→1

x→1

⫽ 4 ⫺ 1 ⫽ 3.

FIGURE 1.57

For x > 1, f 共x兲 is given by 4x ⫺ x2, and you can use direct substitution to obtain lim f 共x兲 ⫽ lim⫹ 共4x ⫺ x2兲

x→1⫹

✓CHECKPOINT 8 Find the limit of f 共x兲 as x approaches 0.



x2 ⫹ 1, f 共x兲 ⫽ 2x ⫹ 1,

x→1

⫽ 4共1兲 ⫺ 12 ⫽ 4 ⫺ 1 ⫽ 3. Because both one-sided limits exist and are equal to 3, it follows that x < 0 x > 0

lim f 共x兲 ⫽ 3.

x→1



The graph in Figure 1.57 confirms this conclusion.

Example 9

Comparing One-Sided Limits

An overnight delivery service charges $12 for the first pound and $2 for each additional pound. Let x represent the weight of a parcel and let f 共x兲 represent the shipping cost.

Shipping cost (in dollars)

y



12, 0 < x ≤ 1 f 共x兲 ⫽ 14, 1 < x ≤ 2 16, 2 < x ≤ 3

Delivery Service Rates

16

Show that the limit of f 共x兲 as x → 2 does not exist.

For 2 < x ≤ 3, f(x) = 16 For 1 < x ≤ 2, f(x) = 14 12 For 0 < x ≤ 1, f(x) = 12 14

The graph of f is shown in Figure 1.58. The limit of f 共x兲 as x approaches 2 from the left is

SOLUTION

10 8

lim f 共x兲 ⫽ 14

6

x→2⫺

4

whereas the limit of f 共x兲 as x approaches 2 from the right is

2 1

2

3

Weight (in pounds)

FIGURE 1.58

Demand Curve

x

lim f 共x兲 ⫽ 16.

x→2⫹

Because these one-sided limits are not equal, the limit of f 共x兲 as x → 2 does not exist.

✓CHECKPOINT 9 Show that the limit of f 共x兲 as x → 1 does not exist in Example 9.



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

90

CHAPTER 1

Functions, Graphs, and Limits

Unbounded Behavior Example 9 shows a limit that fails to exist because the limits from the left and right differ. Another important way in which a limit can fail to exist is when f 共x兲 increases or decreases without bound as x approaches c.

Example 10

Find the limit (if possible).

y

lim

f(x) → ∞ as x → 2+

8 6

x →2

From Figure 1.59, you can see that f 共x兲 decreases without bound as x approaches 2 from the left and f 共x兲 increases without bound as x approaches 2 from the right. Symbolically, you can write this as

2 2

f(x) → −∞ as x → 2−

−4 −6 −8

FIGURE 1.59

4

f(x) =

6

3 x−2

3 x⫺2

SOLUTION

4

−2

An Unbounded Function

8

x

lim⫺

3 ⫽ ⫺⬁ x⫺2

lim

3 ⫽ x⫺2

x→2

and x→2⫹

⬁.

Because f is unbounded as x approaches 2, the limit does not exist.

✓CHECKPOINT 10 Find the limit (if possible):

lim

x→⫺2

5 . x⫹2



STUDY TIP The equal sign in the statement lim⫹ f 共x兲 ⫽ ⬁ does not mean that the limit x→c exists. On the contrary, it tells you how the limit fails to exist by denoting the unbounded behavior of f 共x兲 as x approaches c.

CONCEPT CHECK 1. If limⴚ f 冇x冈 ⴝ limⴙ f 冇x冈, what can you conclude about lim f 冇x冈? x→c

x→c

x→c

2. Describe how to find the limit of a polynomial function p 冇x冈 as x approaches c. 3. Is the limit of f 冇x冈 as x approaches c always equal to f 冇c冈? Why or why not? 4. If f is undefined at x ⴝ c, can you conclude that the limit of f 冇x冈 as x approaches c does not exist? Explain.

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SECTION 1.5

91

Limits

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Section 1.4.

Skills Review 1.5

In Exercises 1– 4, evaluate the expression and simplify. 1. f 共x兲 ⫽ x2 ⫺ 3x ⫹ 3 (a) f 共⫺1兲 2. f 共x兲 ⫽

(b) f 共c兲

冦2x3x ⫺⫹ 2,1,

(a) f 共⫺1兲

(c) f 共x ⫹ h兲

x < 1 x ≥ 1 (b) f 共3兲

(c) f 共t 2 ⫹ 1兲

3. f 共x兲 ⫽ x2 ⫺ 2x ⫹ 2

f 共1 ⫹ h兲 ⫺ f 共1兲 h

4. f 共x兲 ⫽ 4x

f 共2 ⫹ h兲 ⫺ f 共2兲 h

In Exercises 5–8, find the domain and range of the function and sketch its graph. 5. h共x兲 ⫽ ⫺



5 x

6. g共x兲 ⫽ 冪25 ⫺ x2



7. f 共x兲 ⫽ x ⫺ 3

ⱍxⱍ

8. f 共x兲 ⫽

x

In Exercises 9 and 10, determine whether y is a function of x. 9. 9x 2 ⫹ 4y 2 ⫽ 49

10. 2x2 y ⫹ 8x ⫽ 7y

Exercises 1.5

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 8, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. 1. lim 共2x ⫹ 5兲 x→2

x

1.9

1.99

1.999

f 共x兲

2

2.001

2.01

2.1

?

x→2

x⫺2 x2 ⫺ 3x ⫹ 2

x

1.9

4. lim

f 共x兲 5. lim

x→2

x

1.9

1.99

f 共x兲

2

2.001

2.01

2.1

f 共x兲

2.001

2.01

2.1

?

⫺0.1 ⫺0.01

⫺0.001

f 共x兲

0 0.001 0.01

0.1

?

? 6. lim

x⫺2 3. lim 2 x→2 x ⫺ 4 x

2

x

x 1.999

1.999

冪x ⫹ 1 ⫺ 1

x→0

2. lim 共x 2 ⫺ 3x ⫹ 1兲

1.99

1.9

冪x ⫹ 2 ⫺ 冪2

x→0

1.99

1.999

2 ?

2.001

2.01

2.1

x f 共x兲

x

⫺0.1 ⫺0.01

⫺0.001

0 0.001 0.01 ?

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

0.1

92

CHAPTER 1

Functions, Graphs, and Limits

1 1 ⫺ x⫹4 4 7. lim⫺ x→0 x

In Exercises 17–22, use the graph to find the limit (if it exists). (a) limⴙ f 冇x冈

(b) limⴚ f 冇x冈

x→c

x

⫺0.5

⫺0.1

⫺0.01

f 共x兲

y = f(x)

?

0.5

0.1

19. 0.01

0.001

10.

(3, 1)

x

(3, 0)

21.

(−1, 2) x

(a) lim f 共x兲

(b) lim f 共x兲

(b) lim f 共x兲

x→0

(3, −3)

x→1

x→⫺1

y

12.

(−1, 3)

x→⫺1

y = h(x)

x→c

lim g共x兲 ⫽ 9

x→c

x→⫺2

(b) lim h共x兲 x→0

14. lim f 共x兲 ⫽

3 2

lim g共x兲 ⫽

1 2

x→c

In Exercises 15 and 16, find the limit of (a) 冪f 冇x冈, (b) [3f 冇x冈], and (c) [f 冇x冈]2, as x approaches c. 15. lim f 共x兲 ⫽ 16

26. lim 共3x ⫺ 2兲

27. lim 共1 ⫺

28. lim 共⫺x2 ⫹ x ⫺ 2兲

16. lim f 共x兲 ⫽ 9

x→⫺2

x→⫺3

x2



29. lim 冪x ⫹ 6 x→3

x→⫺3

(a) lim h共x兲

x→c

25. lim 共2x ⫹ 5兲

31. lim

(−2, −5)

In Exercises 13 and 14, find the limit of (a) f 冇x冈 1 g冇x冈, (b) f 冇x冈g冇x冈, and (c) f 冇x冈/g冇x冈, as x approaches c. 13. lim f 共x兲 ⫽ 3

24. lim x3

x→1

(0, − 3) x

23. lim x2 x→2

x

(0, 1)

c=3

In Exercises 23– 40, find the limit.

x→3

y

x

(−1, 0)

(a) lim f 共x兲

(b) lim g共x兲

c = −1 y = f (x)

(3, 3)

(0, 1)

x→0

y

22.

y = f(x)

y

(1, − 2)

(a) lim g共x兲

x

y = f(x)

x

x→c

x

y = f(x)

y

(−1, 3)

y = g(x)

c = −2 (−2, 3) (−2, 2)

y = f(x) (3, 0)

y

y

20.

c=3

0

c = −2

(−2, −2)

y

In Exercises 9–12, use the graph to find the limit (if it exists).

11.

y = f(x) x

c=3

?

y = f(x)

y

18.

x

f 共x兲

9.

x→c

(3, 1)

1 1 ⫺ 2⫹x 2 8. lim⫹ x→0 2x x

y

17.

0

⫺0.001

(c) lim f 冇x冈

x→c

2 x⫹2

x2 ⫺ 1 x→⫺2 2x

33. lim 35. lim

x→7

37. lim

x→3

5x x⫹2

x→0 x→2

3 x ⫹ 4 30. lim 冪 x→4

32. lim

3x ⫹ 1 2⫺x

34. lim

4x ⫺ 5 3⫺x

x→⫺2

x→⫺1

36. lim

x→3

冪x ⫹ 1 ⫺ 1

x

38. lim

x→5

冪x ⫹ 1

x⫺4

冪x ⫹ 4 ⫺ 2

x

1 1 ⫺ x⫹4 4 39. lim x→1 x 1 1 ⫺ x⫹2 2 40. lim x→2 x

x→c

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SECTION 1.5 In Exercises 41–60, find the limit (if it exists). 2x ⫺ x ⫺ 3 x⫹1

41. lim

x ⫺1 x⫺1

42. lim

43. lim

x⫺2 x2 ⫺ 4x ⫹ 4

44. lim

2⫺x x2 ⫺ 4

46. lim

t2 ⫹ t ⫺ 2 t2 ⫺ 1

x→1

x→2

2

45. lim

x→⫺1

x→2

t⫹4 ⫺ 16

t→4 t 2

t→1

x3 ⫹ 8 47. lim x→⫺2 x ⫹ 2 49. lim

x→⫺2

2

C⫽

x⫹2

(a) Find the cost of removing 50% of the pollutants. (b) What percent of the pollutants can be removed for $100,000?

51. lim f 共x兲, where f 共x兲 ⫽ x→2

冦40 ⫺ x, 冦

x2 ⫹ 2, 52. lim f 共x兲, where f 共x兲 ⫽ x→1 1,

(c) Evaluate lim ⫺ C. Explain your results. p→100

70. Compound Interest You deposit $2000 in an account that is compounded quarterly at an annual rate of r (in decimal form). The balance A after 10 years is

x⫺2

x→2

x⫽2 x⫽2



A ⫽ 2000 1 ⫹

x⫽1 x⫽1

r 4



40

.

Does the limit of A exist as the interest rate approaches 6%? If so, what is the limit?

1 3

x ⫺ 2, x ≤ 3 冦⫺2x ⫹ 5, x > 3 s, s ≤ 1 54. lim f 共s兲, where f 共s兲 ⫽ 冦 1 ⫺ s, s > 1 53. lim f 共x兲, where f 共x兲 ⫽

25,000p , 0 ≤ p < 100 100 ⫺ p

where C is the cost and p is the percent of pollutants.

ⱍx ⫺ 2ⱍ

50. lim

93

69. Environment The cost (in dollars) of removing p% of the pollutants from the water in a small lake is given by

x3 ⫺ 1 48. lim x→⫺1 x ⫹ 1

ⱍx ⫹ 2ⱍ

Limits

71. Compound Interest Consider a certificate of deposit that pays 10% (annual percentage rate) on an initial deposit of $1000. The balance A after 10 years is

x→3

A ⫽ 1000共1 ⫹ 0.1x兲10兾x

s→1

2共x ⫹ ⌬x兲 ⫺ 2x 55. lim ⌬x→0 ⌬x

where x is the length of the compounding period (in years).

4共x ⫹ ⌬x兲 ⫺ 5 ⫺ 共4x ⫺ 5兲 56. lim ⌬x→0 ⌬x

(b) Use the zoom and trace features to estimate the balance for quarterly compounding and daily compounding.

57. lim

冪x ⫹ 2 ⫹ ⌬x ⫺ 冪x ⫹ 2

(c) Use the zoom and trace features to estimate

⌬x

⌬x→0

58. lim

(a) Use a graphing utility to graph A, where 0 ≤ x ≤ 1.

lim A.

冪x ⫹ ⌬x ⫺ 冪x

x→0 ⫹

⌬x

⌬x→0

What do you think this limit represents? Explain your reasoning.

共t ⫹ ⌬t兲2 ⫺ 5共t ⫹ ⌬t兲 ⫺ 共t2 ⫺ 5t兲 59. lim ⌬t→0 ⌬t 60. lim

⌬t→0

共t ⫹ ⌬t兲 ⫺ 4共t ⫹ ⌬t兲 ⫹ 2 ⫺ 共t ⫺ 4t ⫹ 2兲 ⌬t 2

2

Graphical, Numerical, and Analytic Analysis In Exercises 61–64, use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. 61. lim⫺ x→1

63.

2 x ⫺1

62. lim⫹

5 1⫺x

1 x⫹2

64. lim⫺

x⫹1 x

2

lim ⫺

x→⫺2

x→1

x→0

67. lim

x→⫺4

⫹x⫹4 x ⫹ 2x2 ⫹ 7x ⫺ 4 3

4x2

68. lim

x→⫺2

f 共x兲 ⫽ 共1 ⫹ x兲1兾x is a natural base for many business applications, as you will see in Section 4.2. lim 共1 ⫹ x兲1兾x ⫽ e ⬇ 2.718

(a) Show the reasonableness of this limit by completing the table.

x2 ⫹ 6x ⫺ 7 66. lim 3 x→1 x ⫺ x2 ⫹ 2x ⫺ 2 ⫹ ⫹x⫹6 3x2 ⫺ x ⫺ 14

4x3

73. The limit of

x→0

In Exercises 65–68, use a graphing utility to estimate the limit (if it exists). x2 ⫺ 5x ⫹ 6 65. lim 2 x→2 x ⫺ 4x ⫹ 4

72. Profit Consider the profit function P for the manufacturer in Section 1.4, Exercise 71(b). Does the limit of P exist as x approaches 100? If so, what is the limit?

7x2

x

⫺0.01 ⫺0.001 ⫺0.0001 0 0.0001 0.001 0.01

f 共x兲 (b) Use a graphing utility to graph f and to confirm the answer in part (a). (c) Find the domain and range of the function.

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94

CHAPTER 1

Functions, Graphs, and Limits

Section 1.6 ■ Determine the continuity of functions.

Continuity

■ Determine the continuity of functions on a closed interval. ■ Use the greatest integer function to model and solve real-life problems. ■ Use compound interest models to solve real-life problems.

Continuity In mathematics, the term “continuous” has much the same meaning as it does in everyday use. To say that a function is continuous at x ⫽ c means that there is no interruption in the graph of f at c. The graph of f is unbroken at c, and there are no holes, jumps, or gaps. As simple as this concept may seem, its precise definition eluded mathematicians for many years. In fact, it was not until the early 1800’s that a precise definition was finally developed. Before looking at this definition, consider the function whose graph is shown in Figure 1.60. This figure identifies three values of x at which the function f is not continuous.

y

(c2, f(c2))

1. At x ⫽ c1, f 共c1兲 is not defined. 2. At x ⫽ c2, lim f 共x兲 does not exist. x→c2

(c3, f(c3)) a

c1

c2

c3

b

x

F I G U R E 1 . 6 0 f is not continuous when x ⫽ c1, c2, c3.

3. At x ⫽ c3, f 共c3兲 ⫽ lim f 共x兲. x→c3

At all other points in the interval 共a, b兲, the graph of f is uninterrupted, which implies that the function f is continuous at all other points in the interval 共a, b兲. Definition of Continuity

Let c be a number in the interval 共a, b兲, and let f be a function whose domain contains the interval 共a, b兲. The function f is continuous at the point c if the following conditions are true.

y

1. f 共c兲 is defined. 2. lim f 共x兲 exists. x→c

3. lim f 共x兲 ⫽ f 共c兲. x→c

y = f(x)

a

If f is continuous at every point in the interval 共a, b兲, then it is continuous on an open interval 冇a, b冈. b

F I G U R E 1 . 6 1 On the interval 共a, b兲, the graph of f can be traced with a pencil.

x

Roughly, you can say that a function is continuous on an interval if its graph on the interval can be traced using a pencil and paper without lifting the pencil from the paper, as shown in Figure 1.61.

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SECTION 1.6

TECHNOLOGY Most graphing utilities can draw graphs in two different modes: connected mode and dot mode. The connected mode works well as long as the function is continuous on the entire interval represented by the viewing window. If, however, the function is not continuous at one or more x-values in the viewing window, then the connected mode may try to “connect” parts of the graphs that should not be connected. For instance, try graphing the function y1 ⫽ 共x ⫹ 3兲兾共x ⫺ 2兲 on the viewing window ⫺8 ≤ x ≤ 8 and ⫺6 ≤ y ≤ 6. Do you notice any problems?

In Section 1.5, you studied several types of functions that meet the three conditions for continuity. Specifically, if direct substitution can be used to evaluate the limit of a function at c, then the function is continuous at c. Two types of functions that have this property are polynomial functions and rational functions. Continuity of Polynomial and Rational Functions

1. A polynomial function is continuous at every real number. 2. A rational function is continuous at every number in its domain.

Example 1

Determining Continuity of a Polynomial Function

Discuss the continuity of each function. a. f 共x兲 ⫽ x 2 ⫺ 2x ⫹ 3 b. f 共x兲 ⫽ x 3 ⫺ x Each of these functions is a polynomial function. So, each is continuous on the entire real line, as indicated in Figure 1.62.

SOLUTION

y

y

4

2

3

1

2 1

−2

f(x) = x 2 − 2x + 3

−1

1

2

x3 ⫹ 8 f 共x兲 ⫽ x⫹2 in the standard viewing window. Does the graph appear to be continuous? For what values of x is the function continuous?

FIGURE 1.62

1 −1

3

x

(a)

STUDY TIP A graphing utility can give misleading information about the continuity of a function. Graph the function

95

Continuity

2

x

f(x) = x 3 − x

−2

(b)

Both functions are continuous on 共⫺ ⬁, ⬁兲.

✓CHECKPOINT 1 Discuss the continuity of each function. a. f 共x兲 ⫽ x2 ⫹ x ⫹ 1

b. f 共x兲 ⫽ x3 ⫹ x



Polynomial functions are one of the most important types of functions used in calculus. Be sure you see from Example 1 that the graph of a polynomial function is continuous on the entire real line, and therefore has no holes, jumps, or gaps. Rational functions, on the other hand, need not be continuous on the entire real line, as shown in Example 2.

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96

CHAPTER 1

Functions, Graphs, and Limits

Example 2

Determining Continuity of a Rational Function

Discuss the continuity of each function. a. f 共x兲 ⫽ 1兾x

b. f 共x兲 ⫽ 共x2 ⫺ 1兲兾共x ⫺ 1兲

c. f 共x兲 ⫽ 1兾共x 2 ⫹ 1兲

Each of these functions is a rational function and is therefore continuous at every number in its domain. SOLUTION

a. The domain of f 共x兲 ⫽ 1兾x consists of all real numbers except x ⫽ 0. So, this function is continuous on the intervals 共⫺ ⬁, 0兲 and 共0, ⬁兲. [See Figure 1.63(a).] b. The domain of f 共x兲 ⫽ 共x2 ⫺ 1兲兾共x ⫺ 1兲 consists of all real numbers except x ⫽ 1. So, this function is continuous on the intervals 共⫺ ⬁, 1兲 and 共1, ⬁兲. [See Figure 1.63(b).] c. The domain of f 共x兲 ⫽ 1兾共x2 ⫹ 1兲 consists of all real numbers. So, this function is continuous on the entire real line. [See Figure 1.63(c).] y

y

3

3

3 2

2

f (x) = 1 x

(1, 2)

1

1

−1

y

1

2

3

x

−2

−1

(a) Continuous on 共⫺ ⬁, 0兲 and 共0, ⬁兲.

f(x) = 1

2

f(x) =

x2

−1 x−1

2

3

x

−3

−2

−1

1

−1

−1

−2

−2

(b) Continuous on 共⫺ ⬁, 1兲 and 共1, ⬁兲.

1 x2 + 1

2

x

(c) Continuous on 共⫺ ⬁, ⬁兲.

FIGURE 1.63

✓CHECKPOINT 2 Discuss the continuity of each function. a. f 共x兲 ⫽

1 x⫺1

b. f 共x兲 ⫽

x2 ⫺ 4 x⫺2

c. f 共x兲 ⫽

x2

1 ⫹2



Consider an open interval I that contains a real number c. If a function f is defined on I (except possibly at c), and f is not continuous at c, then f is said to have a discontinuity at c. Discontinuities fall into two categories: removable and nonremovable. A discontinuity at c is called removable if f can be made continuous by appropriately defining (or redefining) f 共c兲. For instance, the function in Example 2(b) has a removable discontinuity at 共1, 2兲. To remove the discontinuity, all you need to do is redefine the function so that f 共1兲 ⫽ 2. A discontinuity at x ⫽ c is nonremovable if the function cannot be made continuous at x ⫽ c by defining or redefining the function at x ⫽ c. For instance, the function in Example 2(a) has a nonremovable discontinuity at x ⫽ 0.

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SECTION 1.6

Continuity

97

Continuity on a Closed Interval The intervals discussed in Examples 1 and 2 are open. To discuss continuity on a closed interval, you can use the concept of one-sided limits, as defined in Section 1.5. Definition of Continuity on a Closed Interval

Let f be defined on a closed interval 关a, b兴. If f is continuous on the open interval 共a, b兲 and lim f 共x兲 ⫽ f 共a兲

x→a ⫹

and

lim f 共x兲 ⫽ f 共b兲

x→b ⫺

then f is continuous on the closed interval [a, b]. Moreover, f is continuous from the right at a and continuous from the left at b. Similar definitions can be made to cover continuity on intervals of the form 共a, b兴 and 关a, b兲, or on infinite intervals. For example, the function f 共x兲 ⫽ 冪x is continuous on the infinite interval 关0, ⬁兲.

Example 3

Examining Continuity at an Endpoint

y

Discuss the continuity of f 共x兲 ⫽ 冪3 ⫺ x.

4

Notice that the domain of f is the set 共⫺ ⬁, 3兴. Moreover, f is continuous from the left at x ⫽ 3 because SOLUTION

3 2

f(x) =

lim f 共x兲 ⫽ lim⫺ 冪3 ⫺ x

3−x

x→3 ⫺

1

−1

FIGURE 1.64

x→3

⫽0 ⫽ f 共3兲. 1

2

3

x

For all x < 3, the function f satisfies the three conditions for continuity. So, you can conclude that f is continuous on the interval 共⫺ ⬁, 3兴, as shown in Figure 1.64.

✓CHECKPOINT 3 Discuss the continuity of f 共x兲 ⫽ 冪x ⫺ 2.



STUDY TIP When working with radical functions of the form f 共x兲 ⫽ 冪g共x兲 remember that the domain of f coincides with the solution of g共x兲 ≥ 0.

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98

CHAPTER 1

Functions, Graphs, and Limits

Example 4

Examining Continuity on a Closed Interval

Discuss the continuity of g共x兲 ⫽

y

冦5x ⫺⫺x,1, 2

⫺1 ≤ x ≤ 2 . 2 < x ≤ 3

The polynomial functions 5 ⫺ x and x2 ⫺ 1 are continuous on the intervals 关⫺1, 2兴 and 共2, 3兴, respectively. So, to conclude that g is continuous on the entire interval 关⫺1, 3兴, you only need to check the behavior of g when x ⫽ 2. You can do this by taking the one-sided limits when x ⫽ 2. SOLUTION

8 7 6

lim g共x兲 ⫽ lim⫺ 共5 ⫺ x兲 ⫽ 3

Limit from the left

lim g共x兲 ⫽ lim⫹ 共x2 ⫺ 1兲 ⫽ 3

Limit from the right

x→2 ⫺

5

x→2

and

4

x→2 ⫹

3

5 − x, −1 ≤ x ≤ 2

g(x) =

2

x→2

Because these two limits are equal,

x 2 − 1, 2 < x ≤ 3

lim g共x兲 ⫽ g共2兲 ⫽ 3.

1

x→2

So, g is continuous at x ⫽ 2 and, consequently, it is continuous on the entire interval 关⫺1, 3兴. The graph of g is shown in Figure 1.65.

x −1

1

2

3

4

FIGURE 1.65

✓CHECKPOINT 4 Discuss the continuity of f 共x兲 ⫽

f(x) = [[x]]

1

−1

−1

1

2

3



x

冀x冁 ⫽ greatest integer less than or equal to x. For example,

−2 −3

FIGURE 1.66 Function

⫺1 ≤ x < 3 . 3 ≤ x ≤ 5

Many functions that are used in business applications are step functions. For instance, the function in Example 9 in Section 1.5 is a step function. The greatest integer function is another example of a step function. This function is denoted by

2

−2

2

The Greatest Integer Function

y

−3

冦x14⫹⫺2,x ,

Greatest Integer

TECHNOLOGY Use a graphing utility to calculate the following. a. 冀3.5冁 b. 冀⫺3.5冁 c. 冀0冁

冀⫺2.1冁 ⫽ greatest integer less than or equal to ⫺2.1 ⫽ ⫺3 冀⫺2冁 ⫽ greatest integer less than or equal to ⫺2 ⫽ ⫺2 冀1.5冁 ⫽ greatest integer less than or equal to 1.5 ⫽ 1. Note that the graph of the greatest integer function (Figure 1.66) jumps up one unit at each integer. This implies that the function is not continuous at each integer. In real-life applications, the domain of the greatest integer function is often restricted to nonnegative values of x. In such cases this function serves the purpose of truncating the decimal portion of x. For example, 1.345 is truncated to 1 and 3.57 is truncated to 3. That is, 冀1.345冁 ⫽ 1

and

冀3.57冁 ⫽ 3.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.6

Example 5

Continuity

99

Modeling a Cost Function

A bookbinding company produces 10,000 books in an eight-hour shift. The fixed cost per shift amounts to $5000, and the unit cost per book is $3. Using the greatest integer function, you can write the cost of producing x books as



C ⫽ 5000 1 ⫹ AP/Wide World Photos

R. R. Donnelley & Sons Company is one of the world’s largest commercial printers. It prints and binds a major share of the national publications in the United States, including Time, Newsweek, and TV Guide.

x⫺1 决10,000 冴冣 ⫹ 3x.

Sketch the graph of this cost function. SOLUTION

Note that during the first eight-hour shift

x⫺1 决10,000 冴 ⫽ 0,

1 ≤ x ≤ 10,000

which implies



C ⫽ 5000 1 ⫹

x⫺1 决10,000 冴冣 ⫹ 3x ⫽ 5000 ⫹ 3x.

During the second eight-hour shift x⫺1 决10,000 冴 ⫽ 1,

10,001 ≤ x ≤ 20,000

which implies



C ⫽ 5000 1 ⫹

x⫺1 决10,000 冴冣 ⫹ 3x

⫽ 10,000 ⫹ 3x. The graph of C is shown in Figure 1.67. Note the graph’s discontinuities. Cost of Producing Books C 110,000 100,000

ird

Cost (in dollars)

90,000

Th

80,000 70,000 60,000

nd

ift

sh

co

Se

50,000 40,000 30,000 20,000

t irs

F

10,000

✓CHECKPOINT 5 Use a graphing utility to graph the cost function in Example 5. ■

ift

sh

ift

sh

x−1 [ ( + 3x ( [ 10,000

C = 5000 1 + 10,000

20,000

30,000

x

Number of books

FIGURE 1.67

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100

CHAPTER 1

Functions, Graphs, and Limits

TECHNOLOGY Step Functions and Compound Functions

To graph a step function or compound function with a graphing utility, you must be familiar with the utility’s programming language. For instance, different graphing utilities have different “integer truncation” functions. One is IPart共x兲, and it yields the truncated integer part of x. For example, IPart共⫺1.2兲 ⫽ ⫺1 and IPart共3.4兲 ⫽ 3. The other function is Int共x兲, which is the greatest integer function. The graphs of these two functions are shown below. When graphing a step function, you should set your graphing utility to dot mode. 2

3

−3

−2

Graph of f 共x兲 ⫽ IPart 共x兲 2

3

−3

−2

Graph of f 共x兲 ⫽ Int 共x兲

On some graphing utilities, you can graph a piecewise-defined function such as f 共x兲 ⫽



x2 ⫺ 4, ⫺x ⫹ 2,

x ≤ 2 . 2 < x

The graph of this function is shown below. 6

−9

9

−6

Consult the user’s guide for your graphing utility for specific keystrokes you can use to graph these functions.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 1.6

101

Extended Application: Compound Interest

TECHNOLOGY You can use a spreadsheet or the table feature of a graphing utility to create a table. Try doing this for the data shown at the right. (Consult the user’s manual of a spreadsheet software program for specific instructions on how to create a table.) Quarterly Compounding A 10,700 10,600

Balance (in dollars)

Continuity

Banks and other financial institutions differ on how interest is paid to an account. If the interest is added to the account so that future interest is paid on previously earned interest, then the interest is said to be compounded. Suppose, for example, that you deposited $10,000 in an account that pays 6% interest, compounded quarterly. Because the 6% is the annual interest rate, the quarterly rate is 14共0.06兲 ⫽ 0.015 or 1.5%. The balances during the first five quarters are shown below. Quarter 1st 2nd 3rd 4th 5th

Balance $10,000.00 10,000.00 10,150.00 10,302.25 10,456.78

⫹ ⫹ ⫹ ⫹

共0.015兲共10,000.00兲 ⫽ $10,150.00 共0.015兲共10,150.00兲 ⫽ $10,302.25 共0.015兲共10,302.25兲 ⫽ $10,456.78 共0.015兲共10,456.78兲 ⫽ $10,613.63

10,500

Example 6

10,400 10,300

Graphing Compound Interest

Sketch the graph of the balance in the account described above.

10,200

Let A represent the balance in the account and let t represent the time, in years. You can use the greatest integer function to represent the balance, as shown.

SOLUTION

10,100 10,000 1 4

1 2

3 4

1

Time (in years)

FIGURE 1.68

5 4

t

A ⫽ 10,000共1 ⫹ 0.015兲冀4t冁 From the graph shown in Figure 1.68, notice that the function has a discontinuity at each quarter.

✓CHECKPOINT 6 Write an equation that gives the balance of the account in Example 6 if the annual interest rate is 8%. ■

CONCEPT CHECK 1. Describe the continuity of a polynomial function. 2. Describe the continuity of a rational function. 3. If a function f is continuous at every point in the interval 冇a, b冈, then what can you say about f on an open interval 冇a, b冈? 4. Describe in your own words what it means to say that a function f is continuous at x ⴝ c.

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102

CHAPTER 1

Skills Review 1.6

Functions, Graphs, and Limits The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 0.4, 0.5, and 1.5.

In Exercises 1– 4, simplify the expression. 1.

x2 ⫹ 6x ⫹ 8 x2 ⫺ 6x ⫺ 16

2.

x2 ⫺ 5x ⫺ 6 x2 ⫺ 9x ⫹ 18

3.

2x2 ⫺ 2x ⫺ 12 4x2 ⫺ 24x ⫹ 36

4.

x3 ⫺ 16x x3 ⫹ 2x2 ⫺ 8x

In Exercises 5–8, solve for x. 5. x2 ⫹ 7x ⫽ 0

6. x2 ⫹ 4x ⫺ 5 ⫽ 0

7. 3x2 ⫹ 8x ⫹ 4 ⫽ 0

8. x3 ⫹ 5x2 ⫺ 24x ⫽ 0

In Exercises 9 and 10, find the limit. 9. lim 共2x2 ⫺ 3x ⫹ 4兲

10. lim 共3x3 ⫺ 8x ⫹ 7兲

x→3

x→⫺2

Exercises 1.6

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–10, determine whether the function is continuous on the entire real line. Explain your reasoning. 1. f 共x兲 ⫽ 5x3 ⫺ x2 ⫹ 2 1 x2 ⫺ 4

4. f 共x兲 ⫽

5. f 共x兲 ⫽

1 4 ⫹ x2

6. f 共x兲 ⫽

9. g共x兲 ⫽

x2

2x ⫺ 1 ⫺ 8x ⫹ 15

8. f 共x兲 ⫽

x 2 ⫺ 4x ⫹ 4 x2 ⫺ 4

10. g共x兲 ⫽

1 9 ⫺ x2

3

3x ⫹1

1

x2

x2

x2 ⫺ 1 x

12. f 共x兲 ⫽

−3 −2 −1

x⫹4 ⫺ 6x ⫹ 5

1 2

3

−3

x −1

−2

−2

−3

−3

3

x

−6

−2

2

6

x

16. f 共x兲 ⫽ 3 ⫺ 2x ⫺ x2 17. f 共x兲 ⫽

x x2 ⫺ 1

18. f 共x兲 ⫽

x⫺3 x2 ⫺ 9

19. f 共x兲 ⫽

x x2 ⫹ 1

20. f 共x兲 ⫽

1 x2 ⫹ 1

21. f 共x兲 ⫽

x⫺5 x2 ⫺ 9x ⫹ 20

22. f 共x兲 ⫽

x⫺1 x2 ⫹ x ⫺ 2

y

1

2

15. f 共x兲 ⫽ x2 ⫺ 2x ⫹ 1

1 x2 ⫺ 4

x

1

−3

2

−1

14 12 10 8 6 2

3

− 3 −2

x3 ⫺ 8 x⫺2 y

2

x 2 ⫺ 9x ⫹ 20 x 2 ⫺ 16

y

14. f 共x兲 ⫽

y

In Exercises 11–34, describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. 11. f 共x兲 ⫽

x2 ⫺ 1 x⫹1

2. f 共x兲 ⫽ 共x2 ⫺ 1兲3

3. f 共x兲 ⫽

7. f 共x兲 ⫽

13. f 共x兲 ⫽

3

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SECTION 1.6 23. f 共x兲 ⫽ 冀2x冁 ⫹ 1

24. f 共x兲 ⫽

冀x冁 ⫹x 2 y

y 3

2

2

−3 − 2

1

2

3

x −2

−1

25. f 共x兲 ⫽

⫹ 3, 冦⫺2x x,



1 2x

x 2 ⫺ 4, 3x ⫹ 1,

2

x

x < 1 x ≥ 1

2

3 ⫹ x, 26. f 共x兲 ⫽ 2 x ⫹ 1,

1 −2

−3

冦3 ⫺⫹x,1,

x ≤ 2 x > 2

28. f 共x兲 ⫽



x ≤ 0 x > 0

42. f 共x兲 ⫽

x⫺3 4x2 ⫺ 12x

冦xx ⫺⫹1,1, x ⫺ 4, 44. f 共x兲 ⫽ 冦 2x ⫹ 4, 2

x < 0 x ≥ 0

2

x ≤ 0 x > 0

In Exercises 45 and 46, find the constant a (Exercise 45) and the constants a and b (Exercise 46) such that the function is continuous on the entire real line. 45. f 共x兲 ⫽

x ≤ 2 x > 2

27. f 共x兲 ⫽

x3 ⫹ x x

43. f 共x兲 ⫽

1

1

41. f 共x兲 ⫽

冦axx , , 3

2

x ≤ 2 x > 2



x ≤ ⫺1 ⫺1 < x < 3 x ≥ 3

2, 46. f 共x兲 ⫽ ax ⫹ b, ⫺2,

In Exercises 47–52, use a graphing utility to graph the function. Use the graph to determine any x-value(s) at which the function is not continuous. Explain why the function is not continuous at the x-value(s).

29. f 共x兲 ⫽

ⱍx ⫹ 1ⱍ

30. f 共x兲 ⫽

ⱍ4 ⫺ xⱍ

47. h共x兲 ⫽

1 x2 ⫺ x ⫺ 2

31. f 共x兲 ⫽ 冀x ⫺ 1冁

48. k 共x兲 ⫽

x⫺4 x2 ⫺ 5x ⫹ 4

x⫹1 4⫺x

32. f 共x兲 ⫽ x ⫺ 冀x冁 1

33. h共x兲 ⫽ f 共g共x兲兲,

f 共x兲 ⫽

34. h共x兲 ⫽ f 共g共x兲兲,

1 f 共x兲 ⫽ , x⫺1

冪x

, g共x兲 ⫽ x ⫺ 1, x > 1 g共x兲 ⫽ x2 ⫹ 5

In Exercises 35–38, discuss the continuity of the function on the closed interval. If there are any discontinuities, determine whether they are removable. Function 35. f 共x兲 ⫽ x ⫺ 4x ⫺ 5 2

Interval

5 x2 ⫹ 1

关⫺2, 2兴

37. f 共x兲 ⫽

1 x⫺2

关1, 4兴

x2 ⫺ 16 x⫺4

x ≤ 3 x > 3

2

x ≤ 1 x > 1

51. f 共x兲 ⫽ x ⫺ 2 冀x冁 52. f 共x兲 ⫽ 冀2x ⫺ 1冁 In Exercises 53–56, describe the interval(s) on which the function is continuous. x x2 ⫹ 1

54. f 共x兲 ⫽ x冪x ⫹ 3 y

y 2

关0, 4兴

40. f 共x兲 ⫽

4

1

−1

In Exercises 39– 44, sketch the graph of the function and describe the interval(s) on which the function is continuous. 39. f 共x兲 ⫽

冦2xx ⫺⫺ 2x,4, 3x ⫺ 1, 50. f 共x兲 ⫽ 冦 x ⫹ 1, 49. f 共x兲 ⫽

53. f 共x兲 ⫽

关⫺1, 5兴

36. f 共x兲 ⫽

x 38. f 共x兲 ⫽ 2 x ⫺ 4x ⫹ 3

103

Continuity

−2

1

2

x

2

(−3, 0) −4

−2

2 −2

2x2 ⫹ x x

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

x

104

CHAPTER 1

1 55. f 共x兲 ⫽ 冀2x冁 2

Functions, Graphs, and Limits 56. f 共x兲 ⫽

y

62. Consumer Awareness The United States Postal Service first class mail rates are $0.41 for the first ounce and $0.17 for each additional ounce or fraction thereof up to 3.5 ounces. A model for the cost C (in dollars) of a first class mailing that weighs 3.5 ounces or less is given below. (Source: United States Postal Service)

y

2

4

1 −3 −2 −1

x⫹1 冪x

1

2

3

x

3 2 1

−2

1

2

3

x

Writing In Exercises 57 and 58, use a graphing utility to graph the function on the interval [ⴚ4, 4]. Does the graph of the function appear to be continuous on this interval? Is the function in fact continuous on [ⴚ4, 4]? Write a short paragraph about the importance of examining a function analytically as well as graphically. x2 ⫹ x 57. f 共x兲 ⫽ x 58. f 共x兲 ⫽

≤ < <
≤ 22 x ⫺ 2, f 共x兲 ⫽ 冦 ⫺x ⫹ 8x ⫺ 14, f 共x兲 ⫽

2

x ≤ 3 x > 3

In Exercises 107–114, describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. 107. f 共x兲 ⫽

1 共x ⫹ 4兲2

108. f 共x兲 ⫽

x⫹2 x

109. f 共x兲 ⫽

3 x⫹1

110. f 共x兲 ⫽

x⫹1 2x ⫹ 2

111. f 共x兲 ⫽ 冀x ⫹ 3冁 112. f 共x兲 ⫽ 冀x冁 ⫺ 2

冦x,x ⫹ 1, xx >≤ 00 x, x ≤ 0 114. f 共x兲 ⫽ 冦 x, x > 0 113. f 共x兲 ⫽

2

In Exercises 115 and 116, find the constant a such that f is continuous on the entire real line. ⫹ 1, 冦⫺x ax ⫺ 8, x ⫹ 1, 116. f 共x兲 ⫽ 冦 2x ⫹ a, 115. f 共x兲 ⫽

x ≤ 3 x > 3 x < 1 x ≥ 1

117. Consumer Awareness The cost C (in dollars) of making x photocopies at a copy shop is given below.



0.15x, 0.10x, C共x兲 ⫽ 0.07x, 0.05x,

0 < t ≤ 1 1 < t ≤ 2 2 < t ≤ 3

where t ⫽ 0 represents the present year. Does the limit of S exist as t approaches 2? Explain your reasoning.

3 x⫽0 104. lim 冪

105. lim f 共x兲 ⫽ 3,

118. Salary Contract A union contract guarantees a 10% salary increase yearly for 3 years. For a current salary of $28,000, the salary S (in thousands of dollars) for the next 3 years is given by

0 < x ≤ 25 25 < x ≤ 100 100 < x ≤ 500 x > 500

(a) Use a graphing utility to graph the function and discuss its continuity. At what values is the function not continuous? Explain your reasoning. (b) Find the cost of making 100 copies.

119. Consumer Awareness A pay-as-you-go cellular phone charges $1 for the first time you access the phone and $0.10 for each additional minute or fraction thereof. Use the greatest integer function to create a model for the cost C of a phone call lasting t minutes. Use a graphing utility to graph the function, and discuss its continuity. 120. Recycling A recycling center pays $0.50 for each pound of aluminum cans. Twenty-four aluminum cans weigh one pound. A mathematical model for the amount A paid by the recycling center is A⫽

1 x 2 24

决 冴

where x is the number of cans. (a) Use a graphing utility to graph the function and then discuss its continuity. (b) How much does the recycling center pay out for 1500 cans? 121. National Debt The table lists the national debt D (in billions of dollars) for selected years. A mathematical model for the national debt is D ⫽ 4.2845t3 ⫺ 97.655t2 ⫹ 861.14t ⫹ 2571.1, 2 ≤ t ≤ 15 where t ⫽ 2 represents 1992. of the Treasury)

(Source: U.S. Department

t

2

3

4

5

6

D

4001.8

4351.0

4643.3

4920.6

5181.5

t

7

8

9

10

11

D

5369.2

5478.2

5605.5

5628.7

5769.9

t

12

13

14

15

D

6198.4

6760.0

7354.7

7905.3

(a) Use a graphing utility to graph the model. (b) Create a table that compares the values given by the model with the actual data. (c) Use the model to estimate the national debt in 2010.

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113

Chapter Test

Chapter Test

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Take this test as you would take a test in class. When you are done, check your work against the answers given in the back of the book. In Exercises 1–3, (a) find the distance between the points, (b) find the midpoint of the line segment joining the points, and (c) find the slope of the line passing through the points. 1. 共1, ⫺1兲, 共⫺4, 4)

2.

共52, 2兲, 共0, 2兲

3. 共3冪2, 2兲, 共冪2, 1兲

4. Sketch the graph of the circle whose general equation is x2 ⫹ y2 ⫺ 4x ⫺ 2y ⫺ 4 ⫽ 0. 5. The demand and supply equations for a product are p ⫽ 65 ⫺ 2.1x and p ⫽ 43 ⫹ 1.9x, respectively, where p is the price in dollars and x represents the number of units in thousands. Find the equilibrium point for this market. In Exercises 6 – 8, find the slope and y-intercept (if possible) of the linear equation. Then sketch the graph of the equation. 6. y ⫽ 15 x ⫺ 2

7. x ⫺ 74 ⫽ 0

8. ⫺x ⫺ 0.4y ⫹ 2.5 ⫽ 0

In Exercises 9–11, (a) graph the function and label the intercepts, (b) determine the domain and range of the function, (c) find the value of the function when x is ⴚ3, ⴚ2, and 3, and (d) determine whether the function is one-to-one. 9. f 共x兲 ⫽ 2x ⫹ 5

10. f 共x兲 ⫽ x2 ⫺ x ⫺ 2

ⱍⱍ

11. f 共x兲 ⫽ x ⫺ 4

In Exercises 12 and 13, find the inverse function of f. Then check your results algebraically by showing that f 冇f ⴚ1冇x冈冈 ⴝ x and f ⴚ1冇f 冇x冈冈 ⴝ x. 12. f 共x兲 ⫽ 4x ⫹ 6

3 8 ⫺ 3x 13. f 共x兲 ⫽ 冪

In Exercises 14–17, find the limit (if it exists). 14. lim

x→0

x⫹5 x⫺5

15. lim

x→5

x⫹5 x⫺5

16. lim

x→⫺3

x2 ⫹ 2x ⫺ 3 x2 ⫹ 4x ⫹ 3

17. lim

冪x ⫹ 9 ⫺ 3

x

x→0

In Exercises 18–20, describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity at a point, identify all conditions of continuity that are not satisfied. 18. f 共x兲 ⫽ t

0

1

2

y

2167

2149

2135

t

3

4

5

y

2127

2113

2101

Table for 21

x2 ⫺ 16 x⫺4

19. f 共x兲 ⫽ 冪5 ⫺ x

20. f 共x兲 ⫽

冦1x ⫺⫺ xx,, 2

x < 1 x ≥ 1

21. The table lists the numbers of farms y (in thousands) in the United States for selected years. A mathematical model for the data is given by y ⫽ 0.54t2 ⫺ 15.4t ⫹ 2166, where t represents the year, with t ⫽ 0 corresponding to 2000. (Source: U.S. Department of Agriculture) (a) Compare the values given by the model with the actual data. How well does the model fit the data? Explain your reasoning. (b) Use the model to predict the number of farms in 2009.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Differentiation

© Schlegelmilch/Corbis

2

2.1 2.2 2.3

2.4 2.5 2.6 2.7 2.8

The Derivative and the Slope of a Graph Some Rules for Differentiation Rates of Change: Velocity and Marginals The Product and Quotient Rules The Chain Rule Higher-Order Derivatives Implicit Differentiation Related Rates

Higher-order derivatives are used to determine the acceleration function of a sports car. The acceleration function shows the changes in the car’s velocity. As the car reaches its “cruising”speed, is the acceleration increasing or decreasing? (See Section 2.6, Exercise 45.)

Applications Differentiation has many real-life applications. The applications listed below represent a sample of the applications in this chapter. ■ ■ ■ ■ ■

Sales, Exercise 61, page 137 Political Fundraiser, Exercise 63, page 137 Make a Decision: Inventory Replenishment, Exercise 65, page 163 Modeling Data, Exercise 51, page 180 Health: U.S. HIV/AIDS Epidemic, Exercise 47, page 187

114 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.1

115

The Derivative and the Slope of a Graph

Section 2.1 ■ Identify tangent lines to a graph at a point.

The Derivative and the Slope of a Graph

■ Approximate the slopes of tangent lines to graphs at points. ■ Use the limit definition to find the slopes of graphs at points. ■ Use the limit definition to find the derivatives of functions. ■ Describe the relationship between differentiability and continuity.

Tangent Line to a Graph y

(x3, y3) (x2, y2)

(x4, y4) x

(x1, y1)

F I G U R E 2 . 1 The slope of a nonlinear graph changes from one point to another.

Calculus is a branch of mathematics that studies rates of change of functions. In this course, you will learn that rates of change have many applications in real life. In Section 1.3, you learned how the slope of a line indicates the rate at which the line rises or falls. For a line, this rate (or slope) is the same at every point on the line. For graphs other than lines, the rate at which the graph rises or falls changes from point to point. For instance, in Figure 2.1, the parabola is rising more quickly at the point 共x1, y1兲 than it is at the point 共x2, y2 兲. At the vertex 共x3, y3兲, the graph levels off, and at the point 共x4, y4兲, the graph is falling. To determine the rate at which a graph rises or falls at a single point, you can find the slope of the tangent line at the point. In simple terms, the tangent line to the graph of a function f at a point P共x1, y1兲 is the line that best approximates the graph at that point, as shown in Figure 2.1. Figure 2.2 shows other examples of tangent lines. y

y

P

y

P y = f (x)

y = f(x)

y = f(x) P

y

x

P(x, y)

FIGURE 2.2

x

FIGURE 2.3 Circle

Tangent Line to a

x

x

Tangent Line to a Graph at a Point

When Isaac Newton (1642–1727) was working on the “tangent line problem,” he realized that it is difficult to define precisely what is meant by a tangent to a general curve. From geometry, you know that a line is tangent to a circle if the line intersects the circle at only one point, as shown in Figure 2.3. Tangent lines to a noncircular graph, however, can intersect the graph at more than one point. For instance, in the second graph in Figure 2.2, if the tangent line were extended, it would intersect the graph at a point other than the point of tangency. In this section, you will see how the notion of a limit can be used to define a general tangent line. D I S C O V E RY Use a graphing utility to graph f 共x兲 ⫽ 2x 3 ⫺ 4x 2 ⫹ 3x ⫺ 5. On the same screen, sketch the graphs of y ⫽ x ⫺ 5, y ⫽ 2x ⫺ 5, and y ⫽ 3x ⫺ 5. Which of these lines, if any, appears to be tangent to the graph of f at the point 共0, ⫺5兲? Explain your reasoning.

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116

CHAPTER 2

Differentiation

Slope of a Graph Because a tangent line approximates the graph at a point, the problem of finding the slope of a graph at a point becomes one of finding the slope of the tangent line at the point. y

Example 1

f (x) = x 2

Approximating the Slope of a Graph

4

Use the graph in Figure 2.4 to approximate the slope of the graph of f 共x兲 ⫽ x 2 at the point 共1, 1兲.

3

From the graph of f 共x兲 ⫽ x 2, you can see that the tangent line at 共1, 1兲 rises approximately two units for each unit change in x. So, the slope of the tangent line at 共1, 1兲 is given by SOLUTION

2

2

Slope ⫽

1

1 1

3

2

x

4

FIGURE 2.4

change in y 2 ⬇ ⫽ 2. change in x 1

Because the tangent line at the point 共1, 1兲 has a slope of about 2, you can conclude that the graph has a slope of about 2 at the point 共1, 1兲.

STUDY TIP When visually approximating the slope of a graph, note that the scales on the horizontal and vertical axes may differ. When this happens (as it frequently does in applications), the slope of the tangent line is distorted, and you must be careful to account for the difference in scales.

✓CHECKPOINT 1 Use the graph to approximate the slope of the graph of f 共x兲 ⫽ x3 at the point 共1, 1兲. y 4

Example 2

3

Figure 2.5 graphically depicts the average monthly temperature (in degrees Fahrenheit) in Duluth, Minnesota. Estimate the slope of this graph at the indicated point and give a physical interpretation of the result. (Source: National

2 1

(1, 1)

−1

1

2

3

Oceanic and Atmospheric Administration)

x

4



Temperature (in degrees Fahrenheit)

From the graph, you can see that the tangent line at the given point falls approximately 28 units for each two-unit change in x. So, you can estimate the slope at the given point to be SOLUTION

change in y ⫺28 ⬇ change in x 2 ⫽ ⫺14 degrees per month.

Slope ⫽

Average Temperature in Duluth y 70 60 50 40 30 20 10

Interpreting Slope

This means that you can expect the average daily temperatures in November to be about 14 degrees lower than the corresponding temperatures in October.

−28° 2 2

4

6

8

10

Month (1 ↔ January)

FIGURE 2.5

12

x

✓CHECKPOINT 2 For which months do the slopes of the tangent lines appear to be positive? Negative? Interpret these slopes in the context of the problem. ■

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.1

The Derivative and the Slope of a Graph

117

Slope and the Limit Process y

In Examples 1 and 2, you approximated the slope of a graph at a point by making a careful graph and then “eyeballing” the tangent line at the point of tangency. A more precise method of approximating the slope of a tangent line makes use of a secant line through the point of tangency and a second point on the graph, as shown in Figure 2.6. If 共x, f 共x兲兲 is the point of tangency and 共x ⫹ ⌬x, f 共x ⫹ ⌬x兲兲 is a second point on the graph of f, then the slope of the secant line through the two points is

(x + ∆ x, f(x + ∆ x))

f (x + ∆ x) − f (x) (x, f(x))

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 . ⌬x

msec ⫽

∆x x

F I G U R E 2 . 6 The Secant Line Through the Two Points 共x, f 共x兲兲 and 共x ⫹ ⌬x, f 共x ⫹ ⌬x兲兲

y

The right side of this equation is called the difference quotient. The denominator ⌬x is the change in x, and the numerator is the change in y. The beauty of this procedure is that you obtain more and more accurate approximations of the slope of the tangent line by choosing points closer and closer to the point of tangency, as shown in Figure 2.7. Using the limit process, you can find the exact slope of the tangent line at 共x, f 共x兲兲, which is also the slope of the graph of f at 共x, f 共x兲兲.

y

(x + ∆ x, f(x + ∆ x))

Slope of secant line

y

y

(x + ∆ x, f(x + ∆ x))

(x + ∆ x, f(x + ∆ x))

∆y (x, f(x))

(x, f (x)) ∆x

FIGURE 2.7

x

(x, f (x))

∆y

(x, f (x)) ∆y

∆x

∆x

x

x

x

As ⌬x approaches 0, the secant lines approach the tangent line.

Definition of the Slope of a Graph

The slope m of the graph of f at the point 共x, f 共x兲兲 is equal to the slope of its tangent line at 共x, f 共x兲兲, and is given by m ⫽ lim msec ⫽ lim ⌬x→0

⌬x→0

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

provided this limit exists.

STUDY TIP ⌬x is used as a variable to represent the change in x in the definition of the slope of a graph. Other variables may also be used. For instance, this definition is sometimes written as m ⫽ lim

h→0

f 共x ⫹ h兲 ⫺ f 共x兲 . h

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

118

CHAPTER 2

Differentiation

Example 3

Algebra Review For help in evaluating the expressions in Examples 3–6, see the review of simplifying fractional expressions on page 196.

Find the slope of the graph of f 共x兲 ⫽ x 2 at the point 共⫺2, 4兲. Begin by finding an expression that represents the slope of a secant line at the point 共⫺2, 4兲. SOLUTION

msec ⫽ ⫽

y

Tangent line at (−2, 4)

5



4



3



2

⫽ f (x) =

1

m = −4 −2

1

Finding Slope by the Limit Process

x2

Set up difference quotient. Use f 共x兲 ⫽ x 2. Expand terms. Simplify. Factor and divide out. Simplify.

Next, take the limit of msec as ⌬x → 0. x

2

f 共⫺2 ⫹ ⌬x兲 ⫺ f 共⫺2兲 ⌬x 共⫺2 ⫹ ⌬x兲2 ⫺ 共⫺2兲2 ⌬x 4 ⫺ 4 ⌬x ⫹ 共⌬x兲2 ⫺ 4 ⌬x ⫺4 ⌬x ⫹ 共⌬x兲2 ⌬x ⌬x共⫺4 ⫹ ⌬x兲 ⌬x ⫺4 ⫹ ⌬x, ⌬x ⫽ 0

m ⫽ lim msec ⫽ lim 共⫺4 ⫹ ⌬x兲 ⫽ ⫺4 ⌬x→0

⌬x→0

So, the graph of f has a slope of ⫺4 at the point 共⫺2, 4兲, as shown in Figure 2.8.

FIGURE 2.8

✓CHECKPOINT 3 Find the slope of the graph of f 共x兲 ⫽ x2 at the point 共2, 4兲.

Example 4



Finding the Slope of a Graph

Find the slope of the graph of f 共x兲 ⫽ ⫺2x ⫹ 4. You know from your study of linear functions that the line given by f 共x兲 ⫽ ⫺2x ⫹ 4 has a slope of ⫺2, as shown in Figure 2.9. This conclusion is consistent with the limit definition of slope.

y

SOLUTION

4

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x→0 ⌬x 关⫺2共x ⫹ ⌬x兲 ⫹ 4兴 ⫺ 关⫺2x ⫹ 4兴 ⫽ lim ⌬x→0 ⌬x

m ⫽ lim

3

f (x) = − 2x + 4 2

⫺2x ⫺ 2 ⌬x ⫹ 4 ⫹ 2x ⫺ 4 ⌬x→0 ⌬x

⫽ lim 1

m = −2

1

FIGURE 2.9

(x, y)

2

⫺2⌬x ⫽ ⫺2 ⌬x→0 ⌬x

⫽ lim 3

x

✓CHECKPOINT 4 Find the slope of the graph of f 共x兲 ⫽ 2x ⫹ 5.



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.1

Use a graphing utility to graph the function y1 ⫽ x 2 ⫹ 1 and the three lines y2 ⫽ 3x ⫺ 1, y3 ⫽ 4x ⫺ 3, and y4 ⫽ 5x ⫺ 5. Which of these lines appears to be tangent to y1 at the point 共2, 5兲? Confirm your answer by showing that the graphs of y1 and its tangent line have only one point of intersection, whereas the graphs of y1 and the other lines each have two points of intersection.

m ⫽ lim

⌬x→0

m ⫽ lim

⌬x→0

Tangent line at (2, 5)

1

2

x

Set up difference quotient. Use f 共x兲 ⫽ x 2 ⫹ 1. Expand terms. Simplify. Factor and divide out. Simplify.

m ⫽ lim msec

Find a formula for the slope of the graph of f 共x兲 ⫽ 4x2 ⫹ 1. What are the slopes at the points 共0, 1兲 and 共1, 5兲? y

−3 − 2 − 1

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x 关共x ⫹ ⌬x兲2 ⫹ 1兴 ⫺ 共x 2 ⫹ 1兲 ⫽ ⌬x 2 ⫹ 2x ⌬x ⫹ 共⌬x兲2 ⫹ 1 ⫺ x 2 ⫺ 1 x ⫽ ⌬x 2 2x ⌬x ⫹ 共⌬x兲 ⫽ ⌬x ⌬x共2x ⫹ ⌬x兲 ⫽ ⌬x ⫽ 2x ⫹ ⌬x, ⌬x ⫽ 0

Next, take the limit of msec as ⌬x → 0.

✓CHECKPOINT 5

1

Finding a Formula for the Slope of a Graph

msec ⫽

FIGURE 2.10

6 5

Formula for slope

SOLUTION

2

−1

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

Find a formula for the slope of the graph of f 共x兲 ⫽ x 2 ⫹ 1. What are the slopes at the points 共⫺1, 2兲 and 共2, 5兲?

f (x) = x 2 + 1

3

−2

Slope at specific point

Except for linear functions, this form will always produce a function of x, which can then be evaluated to find the slope at any desired point.

4

Tangent line at (−1, 2)

f 共c ⫹ ⌬x兲 ⫺ f 共c兲 ⌬x

In Example 4, however, you were finding a formula for the slope at any point on the graph. In such cases, you should use x, rather than c, in the difference quotient.

Example 5

5

119

It is important that you see the distinction between the ways the difference quotients were set up in Examples 3 and 4. In Example 3, you were finding the slope of a graph at a specific point 共c, f 共c兲兲. To find the slope, you can use the following form of a difference quotient.

D I S C O V E RY

y

The Derivative and the Slope of a Graph

⌬x→0

⫽ lim 共2x ⫹ ⌬x兲 ⌬x→0

⫽ 2x Using the formula m ⫽ 2x, you can find the slopes at the specified points. At 共⫺1, 2兲 the slope is m ⫽ 2共⫺1兲 ⫽ ⫺2, and at 共2, 5兲 the slope is m ⫽ 2共2兲 ⫽ 4. The graph of f is shown in Figure 2.10.

(1, 5)

(0, 1) 1 2 3

x

STUDY TIP The slope of the graph of f 共x兲 ⫽ x2 ⫹ 1 varies for different values of x. For what value of x is the slope equal to 0? ■

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120

CHAPTER 2

Differentiation

The Derivative of a Function In Example 5, you started with the function f 共x兲 ⫽ x 2 ⫹ 1 and used the limit process to derive another function, m ⫽ 2x, that represents the slope of the graph of f at the point 共x, f 共x兲兲. This derived function is called the derivative of f at x. It is denoted by f⬘共x兲, which is read as “f prime of x.” STUDY TIP The notation dy兾dx is read as “the derivative of y with respect to x,” and using limit notation, you can write dy ⌬y ⫽ lim ⌬x→0 dx ⌬x ⫽ lim

⌬x→0

⫽ f⬘共x兲.

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

Definition of the Derivative

The derivative of f at x is given by f⬘共x兲 ⫽ lim

⌬x→0

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

provided this limit exists. A function is differentiable at x if its derivative exists at x. The process of finding derivatives is called differentiation. In addition to f⬘共x兲, other notations can be used to denote the derivative of y ⫽ f 共x兲. The most common are dy , dx

d 关 f 共x兲兴, dx

y⬘ ,

Example 6

and

Dx 关 y兴.

Finding a Derivative

Find the derivative of f 共x兲 ⫽ 3x 2 ⫺ 2x. SOLUTION

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x 关3共x ⫹ ⌬x兲2 ⫺ 2共x ⫹ ⌬x兲兴 ⫺ 共3x 2 ⫺ 2x兲 ⫽ lim ⌬x→0 ⌬x 2 ⫹ 6x ⌬x ⫹ 3共⌬x兲2 ⫺ 2x ⫺ 2 ⌬x ⫺ 3x 2 ⫹ 2x 3x ⫽ lim ⌬x→0 ⌬x 2 6x ⌬x ⫹ 3共⌬x兲 ⫺ 2 ⌬x ⫽ lim ⌬x→0 ⌬x ⌬x共6x ⫹ 3 ⌬x ⫺ 2兲 ⫽ lim ⌬x→0 ⌬x ⫽ lim 共6x ⫹ 3 ⌬x ⫺ 2兲

f⬘ 共x兲 ⫽ lim

⌬x→0

⌬x→0

⫽ 6x ⫺ 2 So, the derivative of f 共x兲 ⫽ 3x 2 ⫺ 2x is f⬘共x兲 ⫽ 6x ⫺ 2.

✓CHECKPOINT 6 Find the derivative of f 共x兲 ⫽ x2 ⫺ 5x.



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SECTION 2.1

The Derivative and the Slope of a Graph

121

In many applications, it is convenient to use a variable other than x as the independent variable. Example 7 shows a function that uses t as the independent variable.

Example 7

TECHNOLOGY You can use a graphing utility to confirm the result given in Example 7. One way to do this is to choose a point on the graph of y ⫽ 2兾t, such as 共1, 2兲, and find the equation of the tangent line at that point. Using the derivative found in the example, you know that the slope of the tangent line when t ⫽ 1 is m ⫽ ⫺2. This means that the tangent line at the point 共1, 2兲 is y ⫺ y1 ⫽ m共t ⫺ t1兲 y ⫺ 2 ⫽ ⫺2共t ⫺ 1兲 or

Find the derivative of y with respect to t for the function 2 y⫽ . t Consider y ⫽ f 共t兲, and use the limit process as shown.

SOLUTION

dy f 共t ⫹ ⌬t兲 ⫺ f 共t兲 ⫽ lim ⌬t→0 dt ⌬t 2 2 ⫺ t ⫹ ⌬t t ⫽ lim ⌬t→0 ⌬t 2t ⫺ 2t ⫺ 2 ⌬t t共t ⫹ ⌬t兲 ⫽ lim ⌬t→0 ⌬t

Expand terms. Factor and divide out.

⫽ lim

⫺2 t共t ⫹ ⌬t兲

Simplify.

⌬t→0

⫽⫺

2 t2

Evaluate the limit.

So, the derivative of y with respect to t is dy 2 ⫽ ⫺ 2. dt t

4

6

Remember that the derivative of a function gives you a formula for finding the slope of the tangent line at any point on the graph of the function. For example, the slope of the tangent line to the graph of f at the point 共1, 2兲 is given by f⬘ 共1兲 ⫽ ⫺

−4

Use f 共t兲 ⫽ 2兾t.

⫺2 ⌬t t共⌬t兲共t ⫹ ⌬t兲

⌬t→0

By graphing y ⫽ 2兾t and y ⫽ ⫺2t ⫹ 4 in the same viewing window, as shown below, you can confirm that the line is tangent to the graph at the point 共1, 2兲.*

Set up difference quotient.

⫽ lim

y ⫽ ⫺2t ⫹ 4.

−6

Finding a Derivative

2 ⫽ ⫺2. 12

To find the slopes of the graph at other points, substitute the t-coordinate of the point into the derivative, as shown below. Point

t-Coordinate

Slope

✓CHECKPOINT 7

共2, 1兲

t⫽2

m ⫽ f⬘ 共2兲 ⫽ ⫺

Find the derivative of y with respect to t for the function y ⫽ 4兾t. ■

共⫺2, ⫺1兲

t ⫽ ⫺2

2 1 ⫽⫺ 22 2 2 1 m ⫽ f⬘ 共⫺2兲 ⫽ ⫺ ⫽⫺ 共⫺2兲2 2

*Specific calculator keystroke instructions for operations in this and other technology boxes can be found at college.hmco.com/info/larsonapplied.

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122

CHAPTER 2

Differentiation

Differentiability and Continuity Not every function is differentiable. Figure 2.11 shows some common situations in which a function will not be differentiable at a point—vertical tangent lines, discontinuities, and sharp turns in the graph. Each of the functions shown in Figure 2.11 is differentiable at every value of x except x ⫽ 0. y 2

y

y = x 1/3

1

1

(0, 0) −2

−1

1

2

x −2

Vertical tangent

−1 −2

−1

1

y

y=

x 2/3

2

2

1

1

(0, 0)

1

2

x

−2

FIGURE 2.11

x

y =x

x 1

2

−1

−1 −2

(0, 0)

2

Discontinuity

−2 y

−2

 x y= x

2

Cusp

−2

Node

Functions That Are Not Differentiable at x ⫽ 0

In Figure 2.11, you can see that all but one of the functions are continuous at x ⫽ 0 but none are differentiable there. This shows that continuity is not a strong enough condition to guarantee differentiability. On the other hand, if a function is differentiable at a point, then it must be continuous at that point. This important result is stated in the following theorem. Differentiability Implies Continuity

If a function f is differentiable at x ⫽ c, then f is continuous at x ⫽ c.

CONCEPT CHECK 1. What is the name of the line that best approximates the slope of a graph at a point? 2. What is the name of a line through the point of tangency and a second point on the graph? 3. Sketch a graph of a function whose derivative is always negative. 4. Sketch a graph of a function whose derivative is always positive.

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SECTION 2.1

Skills Review 2.1

123

The Derivative and the Slope of a Graph

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 1.3, 1.4, and 1.5.

In Exercises 1–3, find an equation of the line containing P and Q. 1. P共2, 1兲, Q共2, 4兲

2. P共2, 2兲, Q共⫺5, 2兲

3. P共2, 0兲, Q共3, ⫺1兲

In Exercises 4–7, find the limit. 2x⌬x ⫹ 共⌬x兲2 ⌬x→0 ⌬x

5. lim

3x 2⌬x ⫹ 3x共⌬x兲2 ⫹ 共⌬x兲3 ⌬x→0 ⌬x

1 ⌬x→0 x共x ⫹ ⌬x兲

7. lim

4. lim

共x ⫹ ⌬x兲2 ⫺ x 2 ⌬x→0 ⌬x

6. lim

In Exercises 8 –10, find the domain of the function. 1 x⫺1

8. f 共x兲 ⫽

1 1 9. f 共x兲 ⫽ x3 ⫺ 2x 2 ⫹ x ⫺ 1 5 3

Exercises 2.1

y

2.

(x1, y1)

7.

8. (x, y)

y

(x, y) (x1, y1)

(x2, y2)

(x2, y2) x

3.

6x x3 ⫹ x

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 4, trace the graph and sketch the tangent lines at 冇x1, y1冈 and 冇x2, y2冈. 1.

10. f 共x兲 ⫽

y

4.

x

9.

10. (x, y)

y

(x, y) (x2 , y2)

(x1, y1)

(x1, y1) x

x

In Exercises 5–10, estimate the slope of the graph at the point 冇x, y冈. (Each square on the grid is 1 unit by 1 unit.) 5.

6.

(x, y)

(x, y)

11. Revenue The graph represents the revenue R (in millions of dollars per year) for Polo Ralph Lauren from 1999 through 2005, where t represents the year, with t ⫽ 9 corresponding to 1999. Estimate the slopes of the graph for the years 2002 and 2004. (Source: Polo Ralph Lauren Corp.) Polo Ralph Lauren Revenue R

Revenue (in millions of dollars)

(x 2, y2)

4000 3500 3000 2500 2000 1500 9

10

11

12

13

14

15

t

Year (9 ↔ 1999)

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124

CHAPTER 2

Differentiation

12. Sales The graph represents the sales S (in millions of dollars per year) for Scotts Miracle-Gro Company from 1999 through 2005, where t represents the year, with t ⫽ 9 corresponding to 1999. Estimate the slopes of the graph for the years 2001 and 2004. (Source: Scotts Miracle-Gro Company) Scotts Miracle-Gro Company Sales (in millions of dollars)

In Exercises 15–24, use the limit definition to find the slope of the tangent line to the graph of f at the given point. 16. f 共x兲 ⫽ 2 x ⫹ 4; 共1, 6兲

17. f 共x兲 ⫽ ⫺1; 共0, ⫺1兲

18. f 共x兲 ⫽ 6; 共⫺2, 6兲

2000

19. f 共x兲 ⫽

20. f 共x兲 ⫽ 4 ⫺ x 2; 共2, 0兲

1500

21. f 共x兲 ⫽ x 3 ⫺ x; 共2, 6兲

1000

22. f 共x兲 ⫽ x 3 ⫹ 2 x; 共1, 3兲

2500

500 9

10

11

12

13

14

t

15

Year (9 ↔ 1999)

13. Consumer Trends The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period, where t ⫽ 1 corresponds to January. Estimate the slopes of the graph at t ⫽ 1, 8, and 12.

x2

⫺ 1; 共2, 3兲

23. f 共x兲 ⫽ 2冪x; 共4, 4兲 24. f 共x兲 ⫽ 冪 x ⫹ 1; 共8, 3) In Exercises 25–38, use the limit definition to find the derivative of the function. 25. f 共x兲 ⫽ 3

26. f 共x兲 ⫽ ⫺2

27. f 共x兲 ⫽ ⫺5x

28. f 共x兲 ⫽ 4x ⫹ 1

⫹2

1 30. h共t兲 ⫽ 6 ⫺ 2 t

31. f 共x兲 ⫽ x 2 ⫺ 4

32. f 共x兲 ⫽ 1 ⫺ x 2

1500

33. h共t兲 ⫽ 冪t ⫺ 1

34. f 共x兲 ⫽ 冪x ⫹ 2

1200

35. f 共t兲 ⫽

t3

⫺ 12t

36. f 共t兲 ⫽ t 3 ⫹ t 2

37. f 共x兲 ⫽

1 x⫹2

Visitors to a National Park

Number of visitors (in hundreds of thousands)

(d) Which runner finishes the race first? Explain.

15. f 共x兲 ⫽ 6 ⫺ 2 x; 共2, 2兲

S

V

900 600

29. g共s) ⫽

1 3s

1 s⫺1

38. g共s兲 ⫽

300 1 2 3 4 5 6 7 8 9 10 11 12

t

Month (1 ↔ January)

14. Athletics Two long distance runners starting out side by side begin a 10,000-meter run. Their distances are given by s ⫽ f 共t兲 and s ⫽ g共t兲, where s is measured in thousands of meters and t is measured in minutes. 10,000-Meter Run

Distance (in thousands of meters)

(c) What conclusion can you make regarding their rates at t 3?

s

12 10 8 6 4 2

s = g (t)

s = f (t)

In Exercises 39 – 46, use the limit definition to find an equation of the tangent line to the graph of f at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. 1 39. f 共x兲 ⫽ 2 x 2; 共2, 2兲

40. f 共x兲 ⫽ ⫺x 2; 共⫺1, ⫺1兲

41. f 共x兲 ⫽ 共x ⫺ 1兲2; 共⫺2, 9兲

42. f 共x兲 ⫽ 2x 2 ⫺ 1; 共0, ⫺1兲

43. f 共x兲 ⫽ 冪x ⫹ 1; 共4, 3兲

44. f 共x兲 ⫽ 冪x ⫹ 2; 共7, 3兲

1 45. f 共x兲 ⫽ ; 共1, 1兲 x

46. f 共x兲 ⫽

1 ; 共2, 1兲 x⫺1

In Exercises 47–50, find an equation of the line that is tangent to the graph of f and parallel to the given line. t1 t2 t3

t

Time (in minutes)

(a) Which runner is running faster at t1? (b) What conclusion can you make regarding their rates at t2?

Function

Line

47. f 共x兲 ⫽

⫺ 14x 2

x⫹y⫽0

48. f 共x兲 ⫽

x2

2x ⫹ y ⫽ 0

49. f 共x兲 ⫽

⫺ 12x 3

⫹1

50. f 共x兲 ⫽ x2 ⫺ x

6x ⫹ y ⫹ 4 ⫽ 0 x ⫹ 2y ⫺ 6 ⫽ 0

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SECTION 2.1 In Exercises 51–58, describe the x-values at which the function is differentiable. Explain your reasoning.







51. y ⫽ x ⫹ 3



52. y ⫽ x 2 ⫺ 9 y

y

61. f 共0兲 ⫽ 2; f⬘共x) ⫽ ⫺3, ⫺ ⬁ < x
0 for x > 1

−4 − 2

2

4

Graphical, Numerical, and Analytic Analysis In Exercises 63–66, use a graphing utility to graph f on the interval [ⴚ2, 2]. Complete the table by graphically estimating the slopes of the graph at the given points. Then evaluate the slopes analytically and compare your results with those obtained graphically.

x

6

54. y ⫽ x2兾5 y

y

x

3

4

2 −2

2

−2

4

−3 −2 −1

55. y ⫽ 冪x ⫺ 1

56. y ⫽

1

2

3

x

x2 x ⫺4 2

y

y 5 4 3 2

2 1 1

2

3

x

4

−3

3 4

x

57. y ⫽

冦x

3

x < 0 x ≥ 0

⫹ 3, ⫺ 3,

58. y ⫽

冦⫺x , x 2,

2

x ≤ 1 x > 1

−3 −2

1

2

3

−3 −2 −1

2

−2

−2

−3

−3

3

x

1 x⫺1

60. f 共x兲 ⫽

y

x2 ⫺ 3, 3 ⫺ x 2,

3

3

2

2

1 −2



−1 −2

x ≤ 0 x > 0

y

1

2

3

−2

−1

3 2

2

f⬘ 共x兲 63. f 共x兲 ⫽ 14x 3

64. f 共x兲 ⫽ 12x 2

1 65. f 共x兲 ⫽ ⫺ 2x 3

3 66. f 共x兲 ⫽ ⫺ 2x 2

In Exercises 67–70, find the derivative of the given function f. Then use a graphing utility to graph f and its derivative in the same viewing window. What does the x-intercept of the derivative indicate about the graph of f?

71. The slope of the graph of y ⫽ x 2 is different at every point on the graph of f.

73. If a function is differentiable at a point, then it is continuous at that point. 74. A tangent line to a graph can intersect the graph at more than one point. 75. Writing Use a graphing utility to graph the two functions f 共x兲 ⫽ x 2 ⫹ 1 and g共x兲 ⫽ x ⫹ 1 in the same viewing window. Use the zoom and trace features to analyze the graphs near the point 共0, 1兲. What do you observe? Which function is differentiable at this point? Write a short paragraph describing the geometric significance of differentiability at a point.

ⱍⱍ

1 x

1

72. If a function is continuous at a point, then it is differentiable at that point.

In Exercises 59 and 60, describe the x-values at which f is differentiable. 59. f 共x兲 ⫽

1 2

True or False? In Exercises 71–74, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

1

−1

0

70. f 共x兲 ⫽ x 3 ⫺ 6x 2

2 x

⫺ 12

69. f 共x兲 ⫽ x 3 ⫺ 3x

3 1

⫺1

68. f 共x兲 ⫽ 2 ⫹ 6x ⫺ x 2

y

y

2

⫺ 32

67. f 共x兲 ⫽ x 2 ⫺ 4x

−3

x3

⫺2

f 共x兲

x

6



62. f 共⫺2兲 ⫽ f 共4兲 ⫽ 0; f⬘共1) ⫽ 0, f⬘共x兲 < 0

4 2 −6

In Exercises 61 and 62, identify a function f that has the given characteristics. Then sketch the function.

10

4

125

The Derivative and the Slope of a Graph

1

3

x

−3

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126

CHAPTER 2

Differentiation

Section 2.2 ■ Find the derivatives of functions using the Constant Rule.

Some Rules for Differentiation

■ Find the derivatives of functions using the Power Rule. ■ Find the derivatives of functions using the Constant Multiple Rule. ■ Find the derivatives of functions using the Sum and Difference Rules. ■ Use derivatives to answer questions about real-life situations.

The Constant Rule In Section 2.1, you found derivatives by the limit process. This process is tedious, even for simple functions, but fortunately there are rules that greatly simplify differentiation. These rules allow you to calculate derivatives without the direct use of limits. The Constant Rule

The derivative of a constant function is zero. That is,

y

f (x) = c

d 关c兴 ⫽ 0, dx

The slope of a horizontal line is zero.

PROOF The derivative of a constant function is zero.

FIGURE 2.12

c is a constant.

Let f 共x兲 ⫽ c. Then, by the limit definition of the derivative, you can write

f⬘共x兲 ⫽ lim

⌬x→0

x

So,

f 共x ⫹ ⌬ x兲 ⫺ f 共x兲 c⫺c ⫽ lim ⫽ lim 0 ⫽ 0. ⌬x →0 ⌬x →0 ⌬x ⌬x

d 关c兴 ⫽ 0. dx

STUDY TIP Note in Figure 2.12 that the Constant Rule is equivalent to saying that the slope of a horizontal line is zero.

STUDY TIP An interpretation of the Constant Rule says that the tangent line to a constant function is the function itself. Find an equation of the tangent line to f 共x兲 ⫽ ⫺4 at x ⫽ 3.

Example 1

Finding Derivatives of Constant Functions

d 关7兴 ⫽ 0 dx

b. If f 共x兲 ⫽ 0, then f⬘共x兲 ⫽ 0.

a.

c. If y ⫽ 2, then

dy ⫽ 0. dx

3 d. If g共t兲 ⫽ ⫺ , then g⬘共t兲 ⫽ 0. 2

✓CHECKPOINT 1 Find the derivative of each function. a. f 共x兲 ⫽ ⫺2

b. y ⫽ ␲

c. g共w兲 ⫽ 冪5

d. s共t兲 ⫽ 320.5



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SECTION 2.2

Some Rules for Differentiation

127

The Power Rule The binomial expansion process is used to prove the Power Rule.

共x ⫹ ⌬ x兲2 ⫽ x2 ⫹ 2x ⌬x ⫹ 共⌬x兲2 共x ⫹ ⌬ x兲3 ⫽ x3 ⫹ 3x2 ⌬x ⫹ 3x共⌬x兲2 ⫹ 共⌬x兲3 n共n ⫺ 1兲x n⫺2 共x ⫹ ⌬ x兲n ⫽ xn ⫹ nxn⫺1 ⌬x ⫹ 共⌬x兲2 ⫹ . . . ⫹ 共⌬x兲n 2 共⌬ x兲2 is a factor of these terms.

The (Simple) Power Rule

d n 关x 兴 ⫽ nx n⫺1, dx

n is any real number.

We prove only the case in which n is a positive integer. Let f 共x兲 ⫽ xn. Using the binomial expansion, you can write PROOF

f 共x ⫹ ⌬ x兲 ⫺ f 共x兲 ⌬x 共x ⫹ ⌬ x兲n ⫺ xn ⫽ lim ⌬x→0 ⌬x

f⬘共x兲 ⫽ lim

⌬x→0

Definition of derivative

n共n ⫺ 1兲x n⫺2 共⌬x兲2 ⫹ . . . ⫹ 共⌬x兲n ⫺ x n 2 ⫽ lim ⌬x→0 ⌬x n⫺2 n共n ⫺ 1兲 x ⫽ lim nx n⫺1 ⫹ 共⌬x兲 ⫹ . . . ⫹ 共⌬x兲n⫺1 ⌬x→0 2 ⫽ nxn⫺1 ⫹ 0 ⫹ . . . ⫹ 0 ⫽ nx n⫺1. xn ⫹ nx n⫺1 ⌬x ⫹





For the Power Rule, the case in which n ⫽ 1 is worth remembering as a separate differentiation rule. That is, d 关x兴 ⫽ 1. dx

The derivative of x is 1.

This rule is consistent with the fact that the slope of the line given by y ⫽ x is 1. (See Figure 2.13.) y

y=x 2

∆y 1

∆x

1

FIGURE 2.13

m= 2

∆y =1 ∆x x

The slope of the line y ⫽ x is 1.

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128

CHAPTER 2

Differentiation

Example 2

Applying the Power Rule

Find the derivative of each function. Function Derivative

✓CHECKPOINT 2

a. f 共x兲 ⫽ x3

Find the derivative of each function. 1 a. f 共x兲 ⫽ x 4 b. y ⫽ 3 x

b. y ⫽

1 d. s共t兲 ⫽ t

c. g共w兲 ⫽ w

2

m = −4

d. R ⫽ x 4

dR ⫽ 4x3 dx

3

1

−2

−1

m=2 m=0

1

Rewrite: y ⫽ x⫺2

x

2

Example 3

f 共x兲 ⫽ x2

Finding the Slope of a Graph

Original function

✓CHECKPOINT 3

when x ⫽ ⫺2, ⫺1, 0, 1, and 2.

Find the slopes of the graph of f 共x兲 ⫽ x3 when x ⫽ ⫺1, 0, and 1.

SOLUTION

2 1 1

2

3

x

−2 −3

Derivative

You can use the derivative to find the slopes of the graph of f, as shown.

3

−1

Begin by using the Power Rule to find the derivative of f.

f⬘共x兲 ⫽ 2x

y

−2

Simplify: dy 2 ⫽⫺ 3 dx x

Find the slopes of the graph of

FIGURE 2.14

−3

Differentiate: dy ⫽ 共⫺2兲 x⫺3 dx

Remember that the derivative of a function f is another function that gives the slope of the graph of f at any point at which f is differentiable. So, you can use the derivative to find slopes, as shown in Example 3.

2

m = −2

In Example 2(b), note that before differentiating, you should rewrite 1兾x2 as Rewriting is the first step in many differentiation problems. Original Function: 1 y⫽ 2 x

m=4

dy 2 ⫽ 共⫺2兲x⫺3 ⫽ ⫺ 3 dx x g⬘共t兲 ⫽ 1



f (x) = x 2

4

1 ⫽ x⫺2 x2

c. g共t兲 ⫽ t

x⫺2. y

f⬘共x兲 ⫽ 3x2



x-Value

Slope of Graph of f

x ⫽ ⫺2

m ⫽ f⬘共⫺2兲 ⫽ 2共⫺2兲 ⫽ ⫺4

x ⫽ ⫺1

m ⫽ f⬘共⫺1兲 ⫽ 2共⫺1兲 ⫽ ⫺2

x⫽0

m ⫽ f⬘共0兲 ⫽ 2共0兲 ⫽ 0

x⫽1

m ⫽ f⬘共1兲 ⫽ 2共1兲 ⫽ 2

x⫽2

m ⫽ f⬘共2兲 ⫽ 2共2兲 ⫽ 4

The graph of f is shown in Figure 2.14.

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SECTION 2.2

Some Rules for Differentiation

129

The Constant Multiple Rule To prove the Constant Multiple Rule, the following property of limits is used. lim cg共x兲 ⫽ c 关 lim g共x兲兴

x→a

x→a

The Constant Multiple Rule

If f is a differentiable function of x, and c is a real number, then d 关cf 共x兲兴 ⫽ cf⬘共x兲, dx PROOF

c is a constant.

Apply the definition of the derivative to produce

d cf 共x ⫹ ⌬x兲 ⫺ cf 共x兲 关cf 共x兲兴 ⫽ lim Definition of derivative ⌬x→0 dx ⌬x f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⫽ lim c ⌬x→0 ⌬x f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⫽ c lim ⫽ cf⬘共x兲. ⌬x→0 ⌬x



冥 冥



Informally, the Constant Multiple Rule states that constants can be factored out of the differentiation process. d d 关cf 共x兲兴 ⫽ c 关 dx dx

f 共x兲兴 ⫽ cf⬘共x兲

The usefulness of this rule is often overlooked, especially when the constant appears in the denominator, as shown below. d f 共x兲 d 1 1 d ⫽ 关 f 共x兲 ⫽ dx c dx c c dx

冤 冥









f 共x兲兴 ⫽

1 f⬘共x兲 c

To use the Constant Multiple Rule efficiently, look for constants that can be factored out before differentiating. For example, d d 关5x2兴 ⫽ 5 关x2兴 dx dx ⫽ 5共2x兲 ⫽ 10x

Factor out 5. Differentiate. Simplify.

and d x2 1 d 2 ⫽ 关x 兴 dx 5 5 dx

冤 冥





Factor out 15 .

1 ⫽ 共2x兲 5

Differentiate.

2 ⫽ x. 5

Simplify.

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130

CHAPTER 2

Differentiation

TECHNOLOGY If you have access to a symbolic differentiation utility, try using it to confirm the derivatives shown in this section.

Example 4

Using the Power and Constant Multiple Rules

Differentiate each function. a. y ⫽ 2 x1兾2

b. f 共t兲 ⫽

4t 2 5

SOLUTION

a. Using the Constant Multiple Rule and the Power Rule, you can write dy 1 d 1 d . ⫽ 关2x1兾2兴 ⫽ 2 关x1兾2兴 ⫽ 2 x⫺1兾2 ⫽ x⫺1兾2 ⫽ dx dx dx 2 冪x



Constant Multiple Rule



Power Rule

b. Begin by rewriting f 共t兲 as

✓CHECKPOINT 4 Differentiate each function. a. y ⫽

4 x2

b. f 共x兲 ⫽

16x1兾2



f 共t兲 ⫽

4t 2 4 2 ⫽ t . 5 5

Then, use the Constant Multiple Rule and the Power Rule to obtain f⬘共t兲 ⫽

d 4 2 4 d 2 4 8 共t 兲 ⫽ 共2t兲 ⫽ t. t ⫽ dt 5 5 dt 5 5

冤 冥





You may find it helpful to combine the Constant Multiple Rule and the Power Rule into one combined rule. d 关cxn兴 ⫽ cnx n⫺1, dx

n is a real number, c is a constant.

For instance, in Example 4(b), you can apply this combined rule to obtain d 4 2 4 8 共2兲共t兲 ⫽ t. t ⫽ dt 5 5 5

冤 冥 冢冣

The three functions in the next example are simple, yet errors are frequently made in differentiating functions involving constant multiples of the first power of x. Keep in mind that d 关cx兴 ⫽ c, c is a constant. dx

Example 5

✓CHECKPOINT 5 Find the derivative of each function. a. y ⫽

t 4

b. y ⫽ ⫺

Find the derivative of each function. Original Function a. y ⫽ ⫺

3x 2

b. y ⫽ 3␲x 2x 5

c. y ⫽ ⫺ ■

Applying the Constant Multiple Rule

x 2

Derivative y⬘ ⫽ ⫺

3 2

y⬘ ⫽ 3␲ y⬘ ⫽ ⫺

1 2

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.2

131

Some Rules for Differentiation

Parentheses can play an important role in the use of the Constant Multiple Rule and the Power Rule. In Example 6, be sure you understand the mathematical conventions involving the use of parentheses.

Example 6

Using Parentheses When Differentiating

Find the derivative of each function. a. y ⫽

5 2x3

b. y ⫽

5 共2x兲3

c. y ⫽

7 3x⫺2

d. y ⫽

7 共3x兲⫺2

SOLUTION

Function

Rewrite

Differentiate

Simplify

a. y ⫽

5 2 x3

5 y ⫽ 共x⫺3兲 2

5 y⬘ ⫽ 共⫺3x⫺4兲 2

y⬘ ⫽ ⫺

15 2x4

b. y ⫽

5 共2x兲3

5 y ⫽ 共x⫺3兲 8

5 y⬘ ⫽ 共⫺3x⫺4兲 8

y⬘ ⫽ ⫺

15 8x 4

c. y ⫽

7 3x⫺2

7 y ⫽ 共x2兲 3

7 y⬘ ⫽ 共2x兲 3

y⬘ ⫽

d. y ⫽

7 共3x兲⫺2

y ⫽ 63共x2兲

y⬘ ⫽ 63共2x兲

y⬘ ⫽ 126x

14x 3

✓CHECKPOINT 6 Find the derivative of each function. STUDY TIP When differentiating functions involving radicals, you should rewrite the function with rational exponents. For instance, you 3 x as should rewrite y ⫽ 冪 1兾3 y ⫽ x , and you should rewrite y⫽

1 as y ⫽ x⫺4兾3. 3 冪x 4

a. y ⫽

9 4x2

Example 7

b. y ⫽

9 共4x兲2



Differentiating Radical Functions

Find the derivative of each function. a. y ⫽ 冪x

b. y ⫽

1 3 x2 2冪

c. y ⫽ 冪2 x

SOLUTION

Function

✓CHECKPOINT 7

a. y ⫽ 冪x 1

Find the derivative of each function.

b. y ⫽

a. y ⫽ 冪5x

c. y ⫽ 冪2x

3 x b. y ⫽ 冪



3 x2 2冪

Rewrite

Differentiate

Simplify

y ⫽ x1兾2

y⬘ ⫽

冢12冣 x

1 y ⫽ x⫺2兾3 2

y⬘ ⫽

1 2 ⫺ x⫺5兾3 2 3

y ⫽ 冪2 共x1兾2兲

y⬘ ⫽ 冪2

⫺1兾2

冢 冣

冢12冣 x

⫺1兾2

y⬘ ⫽

1 2冪x

y⬘ ⫽ ⫺ y⬘ ⫽

1 3x5兾3

1 冪2x

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132

CHAPTER 2

Differentiation

The Sum and Difference Rules The next two rules are ones that you might expect to be true, and you may have used them without thinking about it. For instance, if you were asked to differentiate y ⫽ 3x ⫹ 2x3, you would probably write y⬘ ⫽ 3 ⫹ 6x2 without questioning your answer. The validity of differentiating a sum or difference of functions term by term is given by the Sum and Difference Rules. The Sum and Difference Rules

The derivative of the sum or difference of two differentiable functions is the sum or difference of their derivatives. d 关 f 共x) ⫹ g共x兲兴 ⫽ f⬘共x兲 ⫹ g⬘共x兲 dx

Sum Rule

d 关 f 共x兲 ⫺ g共x兲兴 ⫽ f⬘共x兲 ⫺ g⬘共x兲 dx

Difference Rule

PROOF

Let h 共x兲 ⫽ f 共x兲 ⫹ g共x兲. Then, you can prove the Sum Rule as shown. h共x ⫹ ⌬ x兲 ⫺ h共x兲 Definition of derivative ⌬x f 共x ⫹ ⌬ x兲 ⫹ g共x ⫹ ⌬ x兲 ⫺ f 共x兲 ⫺ g共x兲 ⫽ lim ⌬x→0 ⌬x f 共x ⫹ ⌬ x兲 ⫺ f 共x兲 ⫹ g共x ⫹ ⌬ x兲 ⫺ g共x兲 ⫽ lim ⌬x→0 ⌬x f 共x ⫹ ⌬ x兲 ⫺ f 共x兲 g共x ⫹ ⌬ x兲 ⫺ g共x兲 ⫽ lim ⫹ ⌬x→0 ⌬x ⌬x f 共x ⫹ ⌬ x兲 ⫺ f 共x兲 g共x ⫹ ⌬ x兲 ⫺ g共x兲 ⫽ lim ⫹ lim ⌬x→0 ⌬x→0 ⌬x ⌬x ⫽ f⬘共x兲 ⫹ g⬘共x兲

h⬘共x兲 ⫽ lim

⌬x→0





So, d 关 f 共x兲 ⫹ g共x兲兴 ⫽ f⬘共x兲 ⫹ g⬘共x兲. dx The Difference Rule can be proved in a similar manner. The Sum and Difference Rules can be extended to the sum or difference of any finite number of functions. For instance, if y ⫽ f 共x兲 ⫹ g 共x兲 ⫹ h 共x兲, then y⬘ ⫽ f⬘共x兲 ⫹ g⬘共x兲 ⫹ h⬘共x兲. STUDY TIP Look back at Example 6 on page 120. Notice that the example asks for the derivative of the difference of two functions. Verify this result by using the Difference Rule.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.2

With the four differentiation rules listed in this section, you can differentiate any polynomial function.

f (x) = x 3 − 4x + 2 y

Example 8

5

Using the Sum and Difference Rules

Find the slope of the graph of f 共x兲 ⫽ x3 ⫺ 4x ⫹ 2 at the point 共1, ⫺1兲.

4

SOLUTION

The derivative of f 共x兲 is

f⬘共x兲 ⫽ 3x2 ⫺ 4.

2

So, the slope of the graph of f at 共1, ⫺1兲 is

1

−3

133

Some Rules for Differentiation

−1

1 −1

x

2

(1, − 1)

Slope ⫽ f⬘共1兲 ⫽ 3共1兲2 ⫺ 4 ⫽ ⫺1 as shown in Figure 2.15.

✓CHECKPOINT 8

Slope = − 1

Find the slope of the graph of f 共x兲 ⫽ x2 ⫺ 5x ⫹ 1 at the point 共2, ⫺5兲.

FIGURE 2.15



Example 8 illustrates the use of the derivative for determining the shape of a graph. A rough sketch of the graph of f 共x兲 ⫽ x3 ⫺ 4x ⫹ 2 might lead you to think that the point 共1, ⫺1兲 is a minimum point of the graph. After finding the slope at this point to be ⫺1, however, you can conclude that the minimum point (where the slope is 0) is farther to the right. (You will study techniques for finding minimum and maximum points in Section 3.2.) 1

y g(x) = − 2 x 4 + 3x 3 − 2x

Example 9

60

Find an equation of the tangent line to the graph of

50

1 g共x兲 ⫽ ⫺ x 4 ⫹ 3x 3 ⫺ 2x 2

40 30

at the point 共⫺1, ⫺ 32 兲.

20

Slope = 9 −3 −2

− 10 − 20

Using the Sum and Difference Rules

SOLUTION 1

2

3

(− 1, − )

4

5

3 2

FIGURE 2.16

✓CHECKPOINT 9 Find an equation of the tangent line to the graph of f 共x兲 ⫽ ⫺x2 ⫹ 3x ⫺ 2 at the point 共2, 0兲. ■

7

x

The derivative of g共x兲 is g⬘共x兲 ⫽ ⫺2x3 ⫹ 9x2 ⫺ 2, which implies

that the slope of the graph at the point 共⫺1, ⫺ 32 兲 is Slope ⫽ g⬘共⫺1兲 ⫽ ⫺2共⫺1兲3 ⫹ 9共⫺1兲2 ⫺ 2 ⫽2⫹9⫺2 ⫽9

as shown in Figure 2.16. Using the point-slope form, you can write the equation of the tangent line at 共⫺1, ⫺ 32 兲 as shown.

冢 23冣 ⫽ 9关x ⫺ 共⫺1兲兴

y⫺ ⫺

y ⫽ 9x ⫹

15 2

Point-slope form Equation of tangent line

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134

CHAPTER 2

Differentiation

Application Example 10

Modeling Revenue

From 2000 through 2005, the revenue R (in millions of dollars per year) for Microsoft Corporation can be modeled by R ⫽ ⫺110.194t 3 ⫹ 993.98t2 ⫹ 1155.6t ⫹ 23,036,

where t represents the year, with t ⫽ 0 corresponding to 2000. At what rate was Microsoft’s revenue changing in 2001? (Source: Microsoft Corporation)

Microsoft Revenue

One way to answer this question is to find the derivative of the revenue model with respect to time.

Revenue (in millions of dollars)

R

SOLUTION

45,000 40,000 35,000 30,000 25,000 20,000 15,000 10,000 5,000

dR ⫽ ⫺330.582t 2 ⫹ 1987.96t ⫹ 1155.6, 0 ≤ t ≤ 5 dt In 2001 (when t ⫽ 1), the rate of change of the revenue with respect to time is given by

Slope ≈ 2813

⫺330.582共1兲2 ⫹ 1987.96共1兲 ⫹ 1155.6 ⬇ 2813. 1

2

3

4

Year (0 ↔ 2000)

FIGURE 2.17

0 ≤ t ≤ 5

5

t

Because R is measured in millions of dollars and t is measured in years, it follows that the derivative dR兾dt is measured in millions of dollars per year. So, at the end of 2001, Microsoft’s revenue was increasing at a rate of about $2813 million per year, as shown in Figure 2.17.

✓CHECKPOINT 10 From 1998 through 2005, the revenue per share R (in dollars) for McDonald’s Corporation can be modeled by R ⫽ 0.0598t 2 ⫺ 0.379t ⫹ 8.44,

8 ≤ t ≤ 15

where t represents the year, with t ⫽ 8 corresponding to 1998. At what rate was McDonald’s revenue per share changing in 2003? (Source: McDonald’s Corporation) ■

CONCEPT CHECK 1. What is the derivative of any constant function? 2. Write a verbal description of the Power Rule. 3. According to the Sum Rule, the derivative of the sum of two differentiable functions is equal to what? 4. According to the Difference Rule, the derivative of the difference of two differentiable functions is equal to what?

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.2

Skills Review 2.2

135

Some Rules for Differentiation

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 0.3 and 0.4.

In Exercises 1 and 2, evaluate each expression when x ⴝ 2. 1. (a) 2x2 (b) 共2x兲2

2. (a)

(c) 2x⫺2

1 1 共2x兲⫺3 (b) 3 (c) 共3x兲2 4x 4x⫺2

In Exercises 3– 6, simplify the expression. 1 3 4. 2共3兲x2 ⫺ 2x1兾2

3. 4共3兲x3 ⫹ 2共2兲x 5.

共 兲x⫺3兾4

1 1 1 6. 3 共3兲 x2 ⫺ 2共2 兲 x⫺1兾2 ⫹ 3x⫺2兾3

1 4

In Exercises 7–10, solve the equation. 7. 3x2 ⫹ 2x ⫽ 0

8. x3 ⫺ x ⫽ 0

9. x2 ⫹ 8x ⫺ 20 ⫽ 0

10. x2 ⫺ 10x ⫺ 24 ⫽ 0

Exercises 2.2

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 4, find the slope of the tangent line to y ⴝ x n at the point 冇1, 1冈. 1. (a) y ⫽ x2

4. (a) y ⫽ x⫺1兾2

(b) y ⫽ x⫺2 y

y

(b) y ⫽ x1兾2 y

y

(1, 1) (1, 1)

x

x

In Exercises 5– 22, find the derivative of the function.

(b) y ⫽ x3

y

y

5. y ⫽ 3

6. f 共x兲 ⫽ ⫺2

7. y ⫽ x

8. h共x) ⫽ 2x5

4

9. f 共x兲 ⫽ 4x ⫹ 1 11. g共x兲 ⫽ x ⫹ 5x 2

x

x

12. y ⫽ t2 ⫺ 6

14. y ⫽ x 3 ⫺ 9x 2 ⫹ 2 15. s共t兲 ⫽ t 3 ⫺ 2t ⫹ 4 16. y ⫽ 2x3 ⫺ x2 ⫹ 3x ⫺ 1

(b) y ⫽ x⫺1兾3

17. y ⫽ 4t 4兾3

y

y

10. g共x兲 ⫽ 3x ⫺ 1

13. f 共t兲 ⫽ ⫺3t 2 ⫹ 2t ⫺ 4

(1, 1)

(1, 1)

3. (a) y ⫽ x⫺1

x

x

(1, 1)

2. (a) y ⫽ x3兾2

(1, 1)

18. h共x兲 ⫽ x5兾2 19. f 共x兲 ⫽ 4冪x 3 x⫹2 20. g共x兲 ⫽ 4冪

(1, 1)

(1, 1)

21. y ⫽ 4x⫺2 ⫹ 2x2 x

x

22. s共t兲 ⫽ 4t ⫺1 ⫹ 1

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136

CHAPTER 2

Differentiation

In Exercises 23–28, use Example 6 as a model to find the derivative. Function

Rewrite

Differentiate

Simplify

1 23. y ⫽ 3 x







2 24. y ⫽ 2 3x







In Exercises 49–52, (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. Function 49. y ⫽

⫺2x 4

Point ⫹

5x 2

共1, 0兲

⫺3

50. y ⫽ x3 ⫹ x

共⫺1, ⫺2兲

25. y ⫽

1 共4x兲3







26. y ⫽

␲ 共3x兲2







52. f 共x兲 ⫽

冪x

x







4x x⫺3







In Exercises 53–56, determine the point(s), if any, at which the graph of the function has a horizontal tangent line.

27. y ⫽ 28. y ⫽

In Exercises 29–34, find the value of the derivative of the function at the given point. Function

Point

1 x

共1, 1兲

29. f 共x兲 ⫽

30. f 共t兲 ⫽ 4 ⫺ 31. f 共x兲 ⫽

⫺ 12 x 共1

⫹ x 2兲

共1, ⫺1兲





共2, 18兲

2 32. y ⫽ 3x x2 ⫺ x 33. y ⫽ 共2x ⫹ 1兲2

共0, 1兲

34. f 共x兲 ⫽ 3共5 ⫺ x兲2

共5, 0兲



1 3 2 冪 x

5 x 冪

⫺x

共⫺1, 2兲

54. y ⫽ x 3 ⫹ 3x 2

53. y ⫽ ⫺x 4 ⫹ 3x2 ⫺ 1 55. y ⫽

1 2 2x

56. y ⫽ x2 ⫹ 2x

⫹ 5x

In Exercises 57 and 58, (a) sketch the graphs of f and g, (b) find f⬘ 冇1冈 and g⬘ 冇1冈, (c) sketch the tangent line to each graph when x ⴝ 1, and (d) explain the relationship between f⬘ and g⬘. 58. f 共x兲 ⫽ x2

g共x兲 ⫽ x ⫹ 3

g共x兲 ⫽ 3x2

3

59. Use the Constant Rule, the Constant Multiple Rule, and the Sum Rule to find h⬘共1兲 given that f⬘共1兲 ⫽ 3. (a) h 共x兲 ⫽ f 共x兲 ⫺ 2

(b) h共x兲 ⫽ 2f 共x兲 y

y x

In Exercises 35 – 48, find f⬘ 冇x冈. 35. f 共x兲 ⫽ x 2 ⫺

共1, 2兲

57. f 共x兲 ⫽ x3

冢12, 43冣

4 3t

51. f 共x兲 ⫽

3 x 冪

(1, 2)

(1, − 1)

4 ⫺ 3x ⫺2 x

x

36. f 共x兲 ⫽ x2 ⫺ 3x ⫺ 3x⫺2 ⫹ 5x⫺3 37. f 共x兲 ⫽ x2 ⫺ 2x ⫺

2 x4

39. f 共x兲 ⫽ x共x2 ⫹ 1兲

38. f 共x兲 ⫽ x2 ⫹ 4x ⫹

1 x

40. f 共x兲 ⫽ 共x2 ⫹ 2x兲共x ⫹ 1兲

41. f 共x兲 ⫽ 共x ⫹ 4兲共2x 2 ⫺ 1兲

(c) h 共x兲 ⫽ ⫺f 共x兲

y

42. f 共x兲 ⫽ 共3x 2 ⫺ 5x兲共x 2 ⫹ 2兲 2x3 ⫺ 4x2 ⫹ 3 43. f 共x兲 ⫽ x2

2x2 ⫺ 3x ⫹ 1 44. f 共x兲 ⫽ x

4x3 ⫺ 3x 2 ⫹ 2x ⫹ 5 45. f 共x兲 ⫽ x2 46. f 共x兲 ⫽

⫺6x3 ⫹ 3x 2 ⫺ 2x ⫹ 1 x

47. f 共x兲 ⫽ x 4兾5 ⫹ x

(d) h共x兲 ⫽ ⫺1 ⫹ 2 f 共x兲

y

x

(1, − 1)

(1, 1) x

48. f 共x兲 ⫽ x1兾3 ⫺ 1

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.2 60. Revenue The revenue R (in millions of dollars per year) for Polo Ralph Lauren from 1999 through 2005 can be modeled by R ⫽ 0.59221t4 ⫺ 18.0042t3 ⫹ 175.293t2 ⫺ 316.42t ⫺ 116.5 where t is the year, with t ⫽ 9 corresponding to 1999. (Source: Polo Ralph Lauren Corp.) Polo Ralph Lauren Revenue Revenue (in millions of dollars)

R 4000 3500 3000 2500 2000 1500 9

10

11

12

13

14

15

t

137

Some Rules for Differentiation

62. Cost The variable cost for manufacturing an electrical component is $7.75 per unit, and the fixed cost is $500. Write the cost C as a function of x, the number of units produced. Show that the derivative of this cost function is a constant and is equal to the variable cost. 63. Political Fundraiser A politician raises funds by selling tickets to a dinner for $500. The politician pays $150 for each dinner and has fixed costs of $7000 to rent a dining hall and wait staff. Write the profit P as a function of x, the number of dinners sold. Show that the derivative of the profit function is a constant and is equal to the increase in profit from each dinner sold. 64. Psychology: Migraine Prevalence The graph illustrates the prevalence of migraine headaches in males and females in selected income groups. (Source: Adapted from Sue/Sue/Sue, Understanding Abnormal Behavior, Seventh Edition)

Year (9 ↔ 1999)

(a) Find the slopes of the graph for the years 2002 and 2004. (b) Compare your results with those obtained in Exercise 11 in Section 2.1. (c) What are the units for the slope of the graph? Interpret the slope of the graph in the context of the problem. 61. Sales The sales S (in millions of dollars per year) for Scotts Miracle-Gro Company from 1999 through 2005 can be modeled by

Percent of people suffering from migraines

Prevalence of Migraine Headaches 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

Females, < $10,000

Females, ≥ $30,000

Males, < $10,000

Males, ≥ $30,000 10

20

30

40

50

60

70

80

Age

S ⫽ ⫺1.29242t 4 ⫹ 69.9530t3 ⫺ 1364.615t2 ⫹ 11,511.47t ⫺ 33,932.9 where t is the year, with t ⫽ 9 corresponding to 1999. (Source: Scotts Miracle-Gro Company) Scotts Miracle-Gro Company Sales (in millions of dollars)

S 2500 2000 1500

(a) Write a short paragraph describing your general observations about the prevalence of migraines in females and males with respect to age group and income bracket. (b) Describe the graph of the derivative of each curve, and explain the significance of each derivative. Include an explanation of the units of the derivatives, and indicate the time intervals in which the derivatives would be positive and negative. In Exercises 65 and 66, use a graphing utility to graph f and f⬘ over the given interval. Determine any points at which the graph of f has horizontal tangents.

1000 500 9

10

11

12

13

14

15

t

Year (9 ↔ 1999)

(a) Find the slopes of the graph for the years 2001 and 2004. (b) Compare your results with those obtained in Exercise 12 in Section 2.1. (c) What are the units for the slope of the graph? Interpret the slope of the graph in the context of the problem.

Function 65. f 共x兲 ⫽

4.1x 3

Interval ⫺

12x2

⫹ 2.5x

66. f 共x兲 ⫽ x 3 ⫺ 1.4x 2 ⫺ 0.96x ⫹ 1.44

关0, 3兴 关⫺2, 2兴

True or False? In Exercises 67 and 68, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. 67. If f⬘共x兲 ⫽ g⬘共x兲, then f 共x兲 ⫽ g共x兲. 68. If f 共x兲 ⫽ g共x兲 ⫹ c, then f⬘共x兲 ⫽ g⬘共x兲.

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138

CHAPTER 2

Differentiation

Section 2.3 ■ Find the average rates of change of functions over intervals.

Rates of Change: Velocity and Marginals

■ Find the instantaneous rates of change of functions at points. ■ Find the marginal revenues, marginal costs, and marginal profits for

products.

Average Rate of Change In Sections 2.1 and 2.2, you studied the two primary applications of derivatives. 1. Slope The derivative of f is a function that gives the slope of the graph of f at a point 共x, f 共x兲兲. 2. Rate of Change The derivative of f is a function that gives the rate of change of f 共x兲 with respect to x at the point 共x, f 共x兲兲. In this section, you will see that there are many real-life applications of rates of change. A few are velocity, acceleration, population growth rates, unemployment rates, production rates, and water flow rates. Although rates of change often involve change with respect to time, you can investigate the rate of change of one variable with respect to any other related variable. When determining the rate of change of one variable with respect to another, you must be careful to distinguish between average and instantaneous rates of change. The distinction between these two rates of change is comparable to the distinction between the slope of the secant line through two points on a graph and the slope of the tangent line at one point on the graph.

y

(b, f(b))

Definition of Average Rate of Change

If y ⫽ f 共x兲, then the average rate of change of y with respect to x on the interval 关a, b兴 is

f (b) − f (a) (a, f(a))

Average rate of change ⫽ ⫽ a

b b−a

FIGURE 2.18

x

f 共b兲 ⫺ f 共a兲 b⫺a ⌬y . ⌬x

Note that f 共a兲 is the value of the function at the left endpoint of the interval, f 共b兲 is the value of the function at the right endpoint of the interval, and b ⫺ a is the width of the interval, as shown in Figure 2.18.

STUDY TIP In real-life problems, it is important to list the units of measure for a rate of change. The units for ⌬y兾⌬x are “y-units” per “x-units.” For example, if y is measured in miles and x is measured in hours, then ⌬y兾⌬x is measured in miles per hour.

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SECTION 2.3

Example 1 STUDY TIP In Example 1, the average rate of change is positive when the concentration increases and negative when the concentration decreases, as shown in Figure 2.19.

139

Rates of Change: Velocity and Marginals

Medicine

The concentration C (in milligrams per milliliter) of a drug in a patient’s bloodstream is monitored over 10-minute intervals for 2 hours, where t is measured in minutes, as shown in the table. Find the average rate of change over each interval. a. 关0, 10兴

b. 关0, 20兴

c. 关100, 110兴

t

0

10

20

30

40

50

60

70

80

90

100

110

120

C

0

2

17

37

55

73

89

103

111

113

113

103

68

SOLUTION

a. For the interval 关0, 10兴, the average rate of change is Value of C at right endpoint Value of C at left endpoint Drug Concentration in Bloodstream

Concentration (in mg/mL)

C

120 110 100 90 80 70 60 50 40 30 20 10

⌬C 2⫺0 2 ⫽ ⫽ ⫽ 0.2 milligram per milliliter per minute. ⌬t 10 ⫺ 0 10 Width of interval

b. For the interval 关0, 20兴, the average rate of change is ⌬C 17 ⫺ 0 17 ⫽ ⫽ ⫽ 0.85 milligram per milliliter per minute. ⌬t 20 ⫺ 0 20 20

40

60

80 100 120

Time (in minutes)

FIGURE 2.19

t

c. For the interval 关100, 110兴, the average rate of change is ⌬C 103 ⫺ 113 ⫺10 ⫽ ⫽ ⫽ ⫺1 milligram per milliliter per minute. ⌬t 110 ⫺ 100 10

✓CHECKPOINT 1 Use the table in Example 1 to find the average rate of change over each interval. a. 关0, 120兴

b. 关90, 100兴

c. 关90, 120兴



The rates of change in Example 1 are in milligrams per milliliter per minute because the concentration is measured in milligrams per milliliter and the time is measured in minutes. Concentration is measured in milligrams per milliliter. Rate of change is measured in milligrams per milliliter per minute.

⌬C 2⫺0 2 ⫽ ⫽ ⫽ 0.2 milligram per milliliter per minute ⌬t 10 ⫺ 0 10 Time is measured in minutes.

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140

CHAPTER 2

Differentiation

A common application of an average rate of change is to find the average velocity of an object that is moving in a straight line. That is, Average velocity ⫽

change in distance . change in time

This formula is demonstrated in Example 2.

Example 2

Height (in feet)

h 100 90 80 70 60 50 40 30 20 10

t=0 t=1 t = 1.1 t = 1.5 t=2 Falling object

F I G U R E 2 . 2 0 Some falling objects have considerable air resistance. Other falling objects have negligible air resistance. When modeling a falling-body problem, you must decide whether to account for air resistance or neglect it.

Finding an Average Velocity

If a free-falling object is dropped from a height of 100 feet, and air resistance is neglected, the height h (in feet) of the object at time t (in seconds) is given by h ⫽ ⫺16t 2 ⫹ 100.

(See Figure 2.20.)

Find the average velocity of the object over each interval. a. 关1, 2兴

b. 关1, 1.5兴

c. 关1, 1.1兴

You can use the position equation h ⫽ ⫺16t 2 ⫹ 100 to determine the heights at t ⫽ 1, t ⫽ 1.1, t ⫽ 1.5, and t ⫽ 2, as shown in the table. SOLUTION

t (in seconds)

0

1

1.1

1.5

2

h (in feet)

100

84

80.64

64

36

a. For the interval 关1, 2兴, the object falls from a height of 84 feet to a height of 36 feet. So, the average velocity is ⌬h 36 ⫺ 84 ⫺48 ⫽ ⫽ ⫽ ⫺48 feet per second. ⌬t 2⫺1 1 b. For the interval 关1, 1.5兴, the average velocity is ⌬h 64 ⫺ 84 ⫺20 ⫽ ⫽ ⫽ ⫺40 feet per second. ⌬t 1.5 ⫺ 1 0.5 c. For the interval 关1, 1.1兴, the average velocity is ⌬h 80.64 ⫺ 84 ⫺3.36 ⫽ ⫽ ⫽ ⫺33.6 feet per second. ⌬t 1.1 ⫺ 1 0.1

✓CHECKPOINT 2 The height h (in feet) of a free-falling object at time t (in seconds) is given by h ⫽ ⫺16t 2 ⫹ 180. Find the average velocity of the object over each interval. a. 关0, 1兴

b. 关1, 2兴

c. 关2, 3兴



STUDY TIP In Example 2, the average velocities are negative because the object is moving downward.

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SECTION 2.3

141

Rates of Change: Velocity and Marginals

Instantaneous Rate of Change and Velocity Suppose in Example 2 you wanted to find the rate of change of h at the instant t ⫽ 1 second. Such a rate is called an instantaneous rate of change. You can approximate the instantaneous rate of change at t ⫽ 1 by calculating the average rate of change over smaller and smaller intervals of the form 关1, 1 ⫹ ⌬t兴, as shown in the table. From the table, it seems reasonable to conclude that the instantaneous rate of change of the height when t ⫽ 1 is ⫺32 feet per second. ⌬t approaches 0.

⌬t

1

0.5

0.1

0.01

0.001

0.0001

0

⌬h ⌬t

⫺48

⫺40

⫺33.6

⫺32.16

⫺32.016

⫺32.0016

⫺32

⌬h approaches ⫺32. ⌬t

STUDY TIP The limit in this definition is the same as the limit in the definition of the derivative of f at x. This is the second major interpretation of the derivative— as an instantaneous rate of change in one variable with respect to another. Recall that the first interpretation of the derivative is as the slope of the graph of f at x.

Definition of Instantaneous Rate of Change

The instantaneous rate of change (or simply rate of change) of y ⫽ f 共x兲 at x is the limit of the average rate of change on the interval 关x, x ⫹ ⌬x兴, as ⌬x approaches 0. lim

⌬x→0

⌬y f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⫽ lim ⌬x ⌬x→0 ⌬x

If y is a distance and x is time, then the rate of change is a velocity.

Example 3

Finding an Instantaneous Rate of Change

Find the velocity of the object in Example 2 when t ⫽ 1. SOLUTION

From Example 2, you know that the height of the falling object is

given by h ⫽ ⫺16t 2 ⫹ 100.

Position function

By taking the derivative of this position function, you obtain the velocity function. h⬘共t兲 ⫽ ⫺32t

Velocity function

The velocity function gives the velocity at any time. So, when t ⫽ 1, the velocity is h⬘共1兲 ⫽ ⫺32共1兲 ⫽ ⫺32 feet per second.

✓CHECKPOINT 3 Find the velocities of the object in Checkpoint 2 when t ⫽ 1.75 and t ⫽ 2.

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142

CHAPTER 2

Differentiation

D I S C O V E RY Graph the polynomial function h ⫽ ⫺16t 2 ⫹ 16t ⫹ 32 from Example 4 on the domain 0 ≤ t ≤ 2. What is the maximum value of h? What is the derivative of h at this maximum point? In general, discuss how the derivative can be used to find the maximum or minimum values of a function.

The general position function for a free-falling object, neglecting air resistance, is h ⫽ ⫺16t 2 ⫹ v0 t ⫹ h0

Position function

where h is the height (in feet), t is the time (in seconds), v0 is the initial velocity (in feet per second), and h0 is the initial height (in feet). Remember that the model assumes that positive velocities indicate upward motion and negative velocities indicate downward motion. The derivative h⬘ ⫽ ⫺32t ⫹ v0 is the velocity function. The absolute value of the velocity is the speed of the object.

Example 4

Finding the Velocity of a Diver

At time t ⫽ 0, a diver jumps from a diving board that is 32 feet high, as shown in Figure 2.21. Because the diver’s initial velocity is 16 feet per second, his position function is h ⫽ ⫺16t 2 ⫹ 16t ⫹ 32.

Position function

a. When does the diver hit the water? b. What is the diver’s velocity at impact? SOLUTION

a. To find the time at which the diver hits the water, let h ⫽ 0 and solve for t.

32 ft

⫺16t 2 ⫹ 16t ⫹ 32 ⫽ 0 ⫺16共t 2 ⫺ t ⫺ 2兲 ⫽ 0 ⫺16共t ⫹ 1兲共t ⫺ 2兲 ⫽ 0 t ⫽ ⫺1 or t ⫽ 2

Set h equal to 0. Factor out common factor. Factor. Solve for t.

The solution t ⫽ ⫺1 does not make sense in the problem because it would mean the diver hits the water 1 second before he jumps. So, you can conclude that the diver hits the water when t ⫽ 2 seconds. b. The velocity at time t is given by the derivative h⬘ ⫽ ⫺32t ⫹ 16. FIGURE 2.21

Velocity function

The velocity at time t ⫽ 2 is ⫺32共2兲 ⫹ 16 ⫽ ⫺48 feet per second.

✓CHECKPOINT 4 Give the position function of a diver who jumps from a board 12 feet high with initial velocity 16 feet per second. Then find the diver’s velocity function. ■

In Example 4, note that the diver’s initial velocity is v0 ⫽ 16 feet per second (upward) and his initial height is h0 ⫽ 32 feet. Initial velocity is 16 feet per second. Initial height is 32 feet.

h ⫽ ⫺16t 2 ⫹ 16t ⫹ 32

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SECTION 2.3

Rates of Change: Velocity and Marginals

143

Rates of Change in Economics: Marginals Another important use of rates of change is in the field of economics. Economists refer to marginal profit, marginal revenue, and marginal cost as the rates of change of the profit, revenue, and cost with respect to the number x of units produced or sold. An equation that relates these three quantities is P⫽R⫺C where P, R, and C represent the following quantities. P ⫽ total profit R ⫽ total revenue and C ⫽ total cost The derivatives of these quantities are called the marginal profit, marginal revenue, and marginal cost, respectively. dP ⫽ marginal profit dx dR ⫽ marginal revenue dx dC ⫽ marginal cost dx In many business and economics problems, the number of units produced or sold is restricted to positive integer values, as indicated in Figure 2.22(a). (Of course, it could happen that a sale involves half or quarter units, but it is hard to conceive of a sale involving 冪2 units.) The variable that denotes such units is called a discrete variable. To analyze a function of a discrete variable x, you can temporarily assume that x is a continuous variable and is able to take on any real value in a given interval, as indicated in Figure 2.22(b). Then, you can use the methods of calculus to find the x-value that corresponds to the marginal revenue, maximum profit, minimum cost, or whatever is called for. Finally, you should round the solution to the nearest sensible x-value—cents, dollars, units, or days, depending on the context of the problem. y

y 36

36

30

30

24

24

18

18

12

12 6

6 1 2 3 4 5 6 7 8 9 10 11 12

(a) Function of a Discrete Variable

x

1 2 3 4 5 6 7 8 9 10 11 12

(b) Function of a Continuous Variable

FIGURE 2.22

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x

144

CHAPTER 2

Differentiation

Example 5

Finding the Marginal Profit

The profit derived from selling x units of an alarm clock is given by P ⫽ 0.0002x3 ⫹ 10x. a. Find the marginal profit for a production level of 50 units. b. Compare this with the actual gain in profit obtained by increasing the production level from 50 to 51 units. SOLUTION

a. Because the profit is P ⫽ 0.0002x3 ⫹ 10x, the marginal profit is given by the derivative dP兾dx ⫽ 0.0006x 2 ⫹ 10. When x ⫽ 50, the marginal profit is 0.0006共50兲2 ⫹ 10 ⫽ 1.5 ⫹ 10 ⫽ $11.50 per unit.

Marginal profit for x ⫽ 50

b. For x ⫽ 50, the actual profit is Marginal Profit P 600

P ⫽ 共0.0002兲共50兲3 ⫹ 10共50兲 ⫽ 25 ⫹ 500 ⫽ $525.00

(51, 536.53) Marginal profit

(50, 525)

Profit (in dollars)

P ⫽ (0.0002兲共51兲3 ⫹ 10共51兲 ⬇ 26.53 ⫹ 510 ⫽ $536.53.

400 300 200

x 20

30

40

Number of units

FIGURE 2.23

50

Substitute 51 for x. Actual profit for x ⫽ 51

So, the additional profit obtained by increasing the production level from 50 to 51 units is

P = 0.0002x 3 + 10x 10

Actual profit for x ⫽ 50

and for x ⫽ 51, the actual profit is

500

100

Substitute 50 for x.

536.53 ⫺ 525.00 ⫽ $11.53.

Extra profit for one unit

Note that the actual profit increase of $11.53 (when x increases from 50 to 51 units) can be approximated by the marginal profit of $11.50 per unit (when x ⫽ 50), as shown in Figure 2.23.

✓CHECKPOINT 5 Use the profit function in Example 5 to find the marginal profit for a production level of 100 units. Compare this with the actual gain in profit by increasing production from 100 to 101 units. ■ STUDY TIP The reason the marginal profit gives a good approximation of the actual change in profit is that the graph of P is nearly straight over the interval 50 ≤ x ≤ 51. You will study more about the use of marginals to approximate actual changes in Section 3.8.

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SECTION 2.3

Rates of Change: Velocity and Marginals

145

The profit function in Example 5 is unusual in that the profit continues to increase as long as the number of units sold increases. In practice, it is more common to encounter situations in which sales can be increased only by lowering the price per item. Such reductions in price will ultimately cause the profit to decline. The number of units x that consumers are willing to purchase at a given price per unit p is given by the demand function p ⫽ f 共x兲.

Demand function

The total revenue R is then related to the price per unit and the quantity demanded (or sold) by the equation R ⫽ xp.

Revenue function

Example 6

Finding a Demand Function

A business sells 2000 items per month at a price of $10 each. It is estimated that monthly sales will increase 250 units for each $0.25 reduction in price. Use this information to find the demand function and total revenue function. From the given estimate, x increases 250 units each time p drops $0.25 from the original cost of $10. This is described by the equation SOLUTION

x ⫽ 2000 ⫹ 250

⫺p 冢100.25 冣

⫽ 2000 ⫹ 10,000 ⫺ 1000p ⫽ 12,000 ⫺ 1000p. p

Demand Function

Price (in dollars)

14

0.0 R $1 ULA ED REG DUC

12

Solving for p in terms of x produces p ⫽ 12 ⫺

0

RE

$8.75

10

R ⫽ xp

6

2

Demand function

This, in turn, implies that the revenue function is

8

4

x . 1000

p = 12 −

x 1000

3000

6000



x 1000

⫽ 12x ⫺

x2 . 1000

⫽ x 12 ⫺ 9000 12,000

Number of units

FIGURE 2.24

Formula for revenue

x

冣 Revenue function

The graph of the demand function is shown in Figure 2.24. Notice that as the price decreases, the quantity demanded increases.

✓CHECKPOINT 6 Find the demand function in Example 6 if monthly sales increase 200 units for each $0.10 reduction in price. ■

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146

CHAPTER 2

Differentiation

TECHNOLOGY Modeling a Demand Function

To model a demand function, you need data that indicate how many units of a product will sell at a given price. As you might imagine, such data are not easy to obtain for a new product. After a product has been on the market awhile, however, its sales history can provide the necessary data. As an example, consider the two bar graphs shown below. From these graphs, you can see that from 2001 through 2005, the number of prerecorded DVDs sold increased from about 300 million to about 1100 million. During that time, the price per unit dropped from an average price of about $18 to an average price of about $15. (Source: Kagan Research, LLC) Prerecorded DVDs

Prerecorded DVDs

p

1200

Average price per unit (in dollars)

Number of units sold (in millions)

x

1000 800 600 400 200 1

2

3

4

5

t

20 18 16 14 12 10 8 6 4 2 1

Year (1 ↔ 2001)

2

3

4

5

t

Year (1 ↔ 2001)

The information in the two bar graphs is combined in the table, where x represents the units sold (in millions) and p represents the price (in dollars). t

1

2

3

4

5

x

291.5

507.5

713.0

976.6

1072.4

p

18.40

17.11

15.83

15.51

14.94

By entering the ordered pairs 共x, p兲 into a graphing utility, you can find that the power model for the demand for prerecorded DVDs is: p ⫽ 44.55x⫺0.155, 291.5 ≤ x ≤ 1072.4. A graph of this demand function and its data points is shown below 20

200

5

Larson Texts, Inc. • Final Pages • Applied Calculus 8e • CYAN

1100

Short

0.5

Long

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SECTION 2.3

Example 7 p

147

Finding the Marginal Revenue

A fast-food restaurant has determined that the monthly demand for its hamburgers is given by

Demand Function

3.00

Price (in dollars)

Rates of Change: Velocity and Marginals

p⫽

2.50

60,000 ⫺ x . 20,000

Figure 2.25 shows that as the price decreases, the quantity demanded increases. The table shows the demands for hamburgers at various prices.

2.00 1.50 1.00 0.50

p=

60,000 − x 20,000

20,000 40,000 60,000 Number of hamburgers sold

x

F I G U R E 2 . 2 5 As the price decreases, more hamburgers are sold.

x

60,000

50,000

40,000

30,000

20,000

10,000

0

p

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

Find the increase in revenue per hamburger for monthly sales of 20,000 hamburgers. In other words, find the marginal revenue when x ⫽ 20,000. SOLUTION

p⫽

Because the demand is given by

60,000 ⫺ x 20,000

and the revenue is given by R ⫽ xp, you have R ⫽ xp ⫽ x

⫺x 冢60,000 20,000 冣

1 共60,000x ⫺ x 2兲. 20,000 By differentiating, you can find the marginal revenue to be ⫽

dR 1 共60,000 ⫺ 2x兲. ⫽ dx 20,000 So, when x ⫽ 20,000, the marginal revenue is 1 20,000 关60,000 ⫺ 2共20,000兲兴 ⫽ ⫽ $1 per unit. 20,000 20,000

✓CHECKPOINT 7 Find the revenue function and marginal revenue for a demand function of p ⫽ 2000 ⫺ 4x. ■

STUDY TIP Writing a demand function in the form p ⫽ f 共x兲 is a convention used in economics. From a consumer’s point of view, it might seem more reasonable to think that the quantity demanded is a function of the price. Mathematically, however, the two points of view are equivalent because a typical demand function is one-to-one and so has an inverse function. For instance, in Example 7, you could write the demand function as x ⫽ 60,000 ⫺ 20,000p.

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148

CHAPTER 2

Differentiation

Example 8

Finding the Marginal Profit

Suppose in Example 7 that the cost of producing x hamburgers is C ⫽ 5000 ⫹ 0.56x,

0 ≤ x ≤ 50,000.

Find the profit and the marginal profit for each production level. a. x ⫽ 20,000

b. x ⫽ 24,400

c. x ⫽ 30,000

From Example 7, you know that the total revenue from selling x hamburgers is SOLUTION

R⫽

Because the total profit is given by P ⫽ R ⫺ C, you have

Profit Function P

P = 2.44 x −

x 2 − 5000 20,000

P⫽

25,000

1 共60,000x ⫺ x 2兲 ⫺ 共5000 ⫹ 0.56x兲 20,000

⫽ 3x ⫺

20,000 Profit (in dollars)

1 共60,000x ⫺ x 2兲. 20,000

15,000

x2 ⫺ 5000 ⫺ 0.56x 20,000

⫽ 2.44x ⫺

10,000 5,000 20,000 40,000

60,000

x

−5,000 Number of hamburgers sold

FIGURE 2.26

Demand Curve

✓CHECKPOINT 8 From Example 8, compare the marginal profit when 10,000 units are produced with the actual increase in profit from 10,000 units to 10,001 units. ■

x2 ⫺ 5000. 20,000

See Figure 2.26.

So, the marginal profit is dP x ⫽ 2.44 ⫺ . dx 10,000 Using these formulas, you can compute the profit and marginal profit. Production

Profit

Marginal Profit

a. x ⫽ 20,000

P ⫽ $23,800.00

2.44 ⫺

20,000 ⫽ $0.44 per unit 10,000

b. x ⫽ 24,400

P ⫽ $24,768.00

2.44 ⫺

24,400 ⫽ $0.00 per unit 10,000

c. x ⫽ 30,000

P ⫽ $23,200.00

2.44 ⫺

30,000 ⫽ ⫺$0.56 per unit 10,000

CONCEPT CHECK 1. You are asked to find the rate of change of a function over a certain interval. Should you find the average rate of change or the instantaneous rate of change? 2. You are asked to find the rate of change of a function at a certain instant. Should you find the average rate of change or the instantaneous rate of change? 3. If a variable can take on any real value in a given interval, is the variable discrete or continuous? 4. What does a demand function represent?

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SECTION 2.3

Skills Review 2.3

Rates of Change: Velocity and Marginals

149

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 2.1 and 2.2.

In Exercises 1 and 2, evaluate the expression. 1.

⫺63 ⫺ 共⫺105兲 21 ⫺ 7

2.

⫺37 ⫺ 54 16 ⫺ 3

In Exercises 3–10, find the derivative of the function. 3. y ⫽ 4x 2 ⫺ 2x ⫹ 7

4. y ⫽ ⫺3t 3 ⫹ 2t 2 ⫺ 8

5. s ⫽ ⫺16t 2 ⫹ 24t ⫹ 30

9. y ⫽ 12x ⫺

6. y ⫽ ⫺16x 2 ⫹ 54x ⫹ 70 1 8. y ⫽ 9共6x 3 ⫺ 18x 2 ⫹ 63x ⫺ 15兲

⫹ 3r ⫹ 5r兲 2

x2 5000

10. y ⫽ 138 ⫹ 74x ⫺

Exercises 2.3 1. Research and Development The table shows the amounts A (in billions of dollars per year) spent on R&D in the United States from 1980 through 2004, where t is the year, with t ⫽ 0 corresponding to 1980. Approximate the average rate of change of A during each period. (Source: U.S. National Science Foundation) (a) 1980–1985

(b) 1985–1990

(c) 1990–1995

(d) 1995–2000

(e) 1980–2004

(f) 1990–2004

t

0

1

2

3

4

5

6

A

63

72

81

90

102

115

120

t

7

8

9

10

11

12

A

126

134

142

152

161

165

t

13

14

15

16

17

18

A

166

169

184

197

212

228

t

19

20

21

22

23

24

A

245

267

277

276

292

312

2. Trade Deficit The graph shows the values I (in billions of dollars per year) of goods imported to the United States and the values E (in billions of dollars per year) of goods exported from the United States from 1980 through 2005. Approximate each indicated average rate of change. (Source: U.S. International Trade Administration) (a) Imports: 1980–1990

x3 10,000

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

(b) Exports: 1980–1990

(c) Imports: 1990–2000

(d) Exports: 1990–2000

(e) Imports: 1980–2005

(f) Exports: 1980–2005

Trade Deficit 1800

Value of goods (in billions of dollars)

7. A ⫽

1 3 10 共⫺2r

I

1600 1400 1200 1000

E

800 600 400 200 5

10

15

20

25

t

30

Year (0 ↔ 1980) Figure for 2

In Exercises 3–12, use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. 3. f 共t兲 ⫽ 3t ⫹ 5; 关1, 2兴

4. h共x兲 ⫽ 2 ⫺ x; 关0, 2兴

5. h共x兲 ⫽

x2

⫺ 4x ⫹ 2; 关⫺2, 2兴

6. f 共x兲 ⫽

x2

⫺ 6x ⫺ 1; 关⫺1, 3兴

7. f (x) ⫽ 3x4兾3; 关1, 8兴 1 9. f 共x兲 ⫽ ; 关1, 4兴 x

8. f 共x兲 ⫽ x3兾2; 关1, 4] 10. f 共x兲 ⫽

1 冪x

; 关1, 4兴

11. g共x兲 ⫽ x 4 ⫺ x 2 ⫹ 2; 关1, 3兴 12. g共x兲 ⫽ x3 ⫺ 1; 关⫺1, 1兴

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150

CHAPTER 2

Differentiation

13. Consumer Trends The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period, where t ⫽ 1 represents January.

H ⫽ 33共10冪v ⫺ v ⫹ 10.45兲

Number of visitors (in hundreds of thousands)

Visitors to a National Park V

where v is the wind speed (in meters per second).

1500

(a) Find

1200 900

dH and interpret its meaning in this situation. dv

(b) Find the rates of change of H when v ⫽ 2 and when v ⫽ 5.

600 300 1 2 3 4 5 6 7 8 9 10 11 12

t

Month (1 ↔ January)

(a) Estimate the rate of change of V over the interval 关9, 12兴 and explain your results. (b) Over what interval is the average rate of change approximately equal to the rate of change at t ⫽ 8? Explain your reasoning. 14. Medicine The graph shows the estimated number of milligrams of a pain medication M in the bloodstream t hours after a 1000-milligram dose of the drug has been given. Pain Medication in Bloodstream M

Pain medication (in milligrams)

16. Chemistry: Wind Chill At 0⬚ Celsius, the heat loss H (in kilocalories per square meter per hour) from a person’s body can be modeled by

17. Velocity The height s (in feet) at time t (in seconds) of a silver dollar dropped from the top of the Washington Monument is given by s ⫽ ⫺16t 2 ⫹ 555. (a) Find the average velocity on the interval 关2, 3兴. (b) Find the instantaneous velocities when t ⫽ 2 and when t ⫽ 3. (c) How long will it take the dollar to hit the ground? (d) Find the velocity of the dollar when it hits the ground. 18. Physics: Velocity A racecar travels northward on a straight, level track at a constant speed, traveling 0.750 kilometer in 20.0 seconds. The return trip over the same track is made in 25.0 seconds. (a) What is the average velocity of the car in meters per second for the first leg of the run?

1000 800

(b) What is the average velocity for the total trip?

600

(Source: Shipman/Wilson/Todd, An Introduction to Physical Science, Eleventh Edition)

400 200 1

2

3

4

5

6

7

t

Hours

Marginal Cost In Exercises 19–22, find the marginal cost for producing x units. (The cost is measured in dollars.)

(a) Estimate the one-hour interval over which the average rate of change is the greatest.

19. C ⫽ 4500 ⫹ 1.47x

(b) Over what interval is the average rate of change approximately equal to the rate of change at t ⫽ 4? Explain your reasoning.

22. C ⫽ 100共9 ⫹ 3冪x 兲

15. Medicine The effectiveness E (on a scale from 0 to 1) of a pain-killing drug t hours after entering the bloodstream is given by 1 E ⫽ 共9t ⫹ 3t 2 ⫺ t 3兲, 27

0 ≤ t ≤ 4.5.

21. C ⫽ 55,000 ⫹ 470x ⫺

20. C ⫽ 205,000 ⫹ 9800x 0.25x 2,

0 ≤ x ≤ 940

Marginal Revenue In Exercises 23–26, find the marginal revenue for producing x units. (The revenue is measured in dollars.) 23. R ⫽ 50x ⫺ 0.5x 2 25. R ⫽

⫺6x 3



8x 2

24. R ⫽ 30x ⫺ x 2 ⫹ 200x

26. R ⫽ 50共20x ⫺ x3兾2兲

Find the average rate of change of E on each indicated interval and compare this rate with the instantaneous rates of change at the endpoints of the interval.

Marginal Profit In Exercises 27–30, find the marginal profit for producing x units. (The profit is measured in dollars.)

(a) 关0, 1兴

27. P ⫽ ⫺2x 2 ⫹ 72x ⫺ 145

(b) 关1, 2兴

(c) 关2, 3兴

(d) 关3, 4兴

28. P ⫽ ⫺0.25x 2 ⫹ 2000x ⫺ 1,250,000 29. P ⫽ ⫺0.00025x 2 ⫹ 12.2x ⫺ 25,000 30. P ⫽ ⫺0.5x 3 ⫹ 30x 2 ⫺ 164.25x ⫺ 1000

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SECTION 2.3 31. Marginal Cost The cost C (in dollars) of producing x units of a product is given by C ⫽ 3.6冪x ⫹ 500. (a) Find the additional cost when the production increases from 9 to 10 units. (b) Find the marginal cost when x ⫽ 9. (c) Compare the results of parts (a) and (b). 32. Marginal Revenue The revenue R (in dollars) from renting x apartments can be modeled by R ⫽ 2x共900 ⫹ 32x ⫺ x 2兲. (a) Find the additional revenue when the number of rentals is increased from 14 to 15. (b) Find the marginal revenue when x ⫽ 14. (c) Compare the results of parts (a) and (b). 33. Marginal Profit The profit P (in dollars) from selling x units of calculus textbooks is given by P ⫽ ⫺0.05x 2 ⫹ 20x ⫺ 1000. (a) Find the additional profit when the sales increase from 150 to 151 units. (b) Find the marginal profit when x ⫽ 150. (c) Compare the results of parts (a) and (b). 34. Population Growth The population P (in thousands) of Japan can be modeled by P ⫽ ⫺14.71t2 ⫹ 785.5t ⫹ 117,216 where t is time in years, with t ⫽ 0 corresponding to 1980. (Source: U.S. Census Bureau) (a) Evaluate P for t ⫽ 0, 10, 15, 20, and 25. Explain these values. (b) Determine the population growth rate, dP兾dt. (c) Evaluate dP兾dt for the same values as in part (a). Explain your results. 35. Health The temperature T (in degrees Fahrenheit) of a person during an illness can be modeled by the equation T ⫽ ⫺0.0375t 2 ⫹ 0.3t ⫹ 100.4, where t is time in hours since the person started to show signs of a fever. (a) Use a graphing utility to graph the function. Be sure to choose an appropriate window. (b) Do the slopes of the tangent lines appear to be positive or negative? What does this tell you? (c) Evaluate the function for t ⫽ 0, 4, 8, and 12. (d) Find dT兾dt and explain its meaning in this situation. (e) Evaluate dT兾dt for t ⫽ 0, 4, 8, and 12. 36. Marginal Profit The profit P (in dollars) from selling x units of a product is given by P ⫽ 36,000 ⫹ 2048冪x ⫺

1 , 8x2

150 ≤ x ≤ 275.

151

Rates of Change: Velocity and Marginals

Find the marginal profit for each of the following sales. (a) x ⫽ 150

(b) x ⫽ 175

(c) x ⫽ 200

(d) x ⫽ 225

(e) x ⫽ 250

(f) x ⫽ 275

37. Profit The monthly demand function and cost function for x newspapers at a newsstand are given by p ⫽ 5 ⫺ 0.001x and C ⫽ 35 ⫹ 1.5x. (a) Find the monthly revenue R as a function of x. (b) Find the monthly profit P as a function of x. (c) Complete the table. x

600

1200

1800

2400

3000

dR兾dx dP兾dx P 38. Economics

Use the table to answer the questions below.

Quantity produced and sold (Q)

Price (p)

Total revenue (TR)

Marginal revenue (MR)

0 2 4 6 8 10

160 140 120 100 80 60

0 280 480 600 640 600

— 130 90 50 10 ⫺30

(a) Use the regression feature of a graphing utility to find a quadratic model that relates the total revenue 共TR兲 to the quantity produced and sold 共Q兲. (b) Using derivatives, find a model for marginal revenue from the model you found in part (a). (c) Calculate the marginal revenue for all values of Q using your model in part (b), and compare these values with the actual values given. How good is your model? (Source: Adapted from Taylor, Economics, Fifth Edition) 39. Marginal Profit When the price of a glass of lemonade at a lemonade stand was $1.75, 400 glasses were sold. When the price was lowered to $1.50, 500 glasses were sold. Assume that the demand function is linear and that the variable and fixed costs are $0.10 and $25, respectively. (a) Find the profit P as a function of x, the number of glasses of lemonade sold. (b) Use a graphing utility to graph P, and comment about the slopes of P when x ⫽ 300 and when x ⫽ 700. (c) Find the marginal profits when 300 glasses of lemonade are sold and when 700 glasses of lemonade are sold.

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152

CHAPTER 2

Differentiation

40. Marginal Cost The cost C of producing x units is modeled by C ⫽ v共x兲 ⫹ k, where v represents the variable cost and k represents the fixed cost. Show that the marginal cost is independent of the fixed cost.

46. Gasoline Sales The number N of gallons of regular unleaded gasoline sold by a gasoline station at a price of p dollars per gallon is given by N ⫽ f 共p兲.

41. Marginal Profit When the admission price for a baseball game was $6 per ticket, 36,000 tickets were sold. When the price was raised to $7, only 33,000 tickets were sold. Assume that the demand function is linear and that the variable and fixed costs for the ballpark owners are $0.20 and $85,000, respectively.

(b) Is f⬘共2.959) usually positive or negative? Explain.

(a) Find the profit P as a function of x, the number of tickets sold. (b) Use a graphing utility to graph P, and comment about the slopes of P when x ⫽ 18,000 and when x ⫽ 36,000. (c) Find the marginal profits when 18,000 tickets are sold and when 36,000 tickets are sold. 42. Marginal Profit In Exercise 41, suppose ticket sales decreased to 30,000 when the price increased to $7. How would this change the answers? 43. Profit The demand function for a product is given by p ⫽ 50兾冪x for 1 ≤ x ≤ 8000, and the cost function is given by C ⫽ 0.5x ⫹ 500 for 0 ≤ x ≤ 8000. Find the marginal profits for (a) x ⫽ 900, (b) x ⫽ 1600, (c) x ⫽ 2500, and (d) x ⫽ 3600.

(a) Describe the meaning of f⬘共2.959) 47. Dow Jones Industrial Average The table shows the year-end closing prices p of the Dow Jones Industrial Average (DJIA) from 1992 through 2006, where t is the year, and t ⫽ 2 corresponds to 1992. (Source: Dow Jones Industrial Average) t

2

p

3

4

5

6

3301.11 3754.09

3834.44

5117.12

6448.26

t

7

9

10

11

p

7908.24 9181.43

11,497.12 10,786.85 10,021.50

t

12

14

p

8341.63 10,453.92 10,783.01 10,717.50 12,463.15

8

13

15

16

(a) Determine the average rate of change in the value of the DJIA from 1992 to 2006.

If you were in charge of setting the price for this product, what price would you set? Explain your reasoning.

(b) Estimate the instantaneous rate of change in 1998 by finding the average rate of change from 1996 to 2000.

44. Inventory Management The annual inventory cost for a manufacturer is given by

(c) Estimate the instantaneous rate of change in 1998 by finding the average rate of change from 1997 to 1999.

C ⫽ 1,008,000兾Q ⫹ 6.3Q where Q is the order size when the inventory is replenished. Find the change in annual cost when Q is increased from 350 to 351, and compare this with the instantaneous rate of change when Q ⫽ 350. 45. MAKE A DECISION: FUEL COST A car is driven 15,000 miles a year and gets x miles per gallon. Assume that the average fuel cost is $2.95 per gallon. Find the annual cost of fuel C as a function of x and use this function to complete the table. 10

15

20

25

30

35

48. Biology Many populations in nature exhibit logistic growth, which consists of four phases, as shown in the figure. Describe the rate of growth of the population in each phase, and give possible reasons as to why the rates might be changing from phase to phase. (Source: Adapted from Levine/Miller, Biology: Discovering Life, Second Edition) Acceleration Deceleration phase phase

40

C dC兾dx Who would benefit more from a 1 mile per gallon increase in fuel efficiency—the driver who gets 15 miles per gallon or the driver who gets 35 miles per gallon? Explain.

Lag phase

Population

x

(d) Compare your answers for parts (b) and (c). Which interval do you think produced the best estimate for the instantaneous rate of change in 1998?

Equilibrium

Time

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SECTION 2.4

153

The Product and Quotient Rules

Section 2.4

The Product and Quotient Rules

■ Find the derivatives of functions using the Product Rule. ■ Find the derivatives of functions using the Quotient Rule. ■ Simplify derivatives. ■ Use derivatives to answer questions about real-life situations.

The Product Rule In Section 2.2, you saw that the derivative of a sum or difference of two functions is simply the sum or difference of their derivatives. The rules for the derivative of a product or quotient of two functions are not as simple. STUDY TIP Rather than trying to remember the formula for the Product Rule, it can be more helpful to remember its verbal statement: the first function times the derivative of the second plus the second function times the derivative of the first.

The Product Rule

The derivative of the product of two differentiable functions is equal to the first function times the derivative of the second plus the second function times the derivative of the first. d 关 f 共x兲g共x兲兴 ⫽ f 共x兲g⬘共x兲 ⫹ g共x兲f⬘共x兲 dx PROOF Some mathematical proofs, such as the proof of the Sum Rule, are straightforward. Others involve clever steps that may not appear to follow clearly from a prior step. The proof below involves such a step—adding and subtracting the same quantity. (This step is shown in color.) Let F共x兲 ⫽ f 共x兲g共x兲.

F共x ⫹ ⌬x兲 ⫺ F共x兲 ⌬x f 共x ⫹ ⌬x兲g共x ⫹ ⌬x兲 ⫺ f 共x兲g共x兲 ⫽ lim ⌬x→0 ⌬x f 共x ⫹ ⌬x兲g共x ⫹ ⌬x兲 ⫺ f 共x ⫹ ⌬x兲g共x兲 ⫹ f 共x ⫹ ⌬ x兲g共x兲 ⫺ f 共x兲g共x兲 ⫽ lim ⌬x→0 ⌬x g共x ⫹ ⌬ x兲 ⫺ g共x兲 f 共x ⫹ ⌬ x兲 ⫺ f 共x兲 ⫽ lim f 共x ⫹ ⌬x兲 ⫹ g共x兲 ⌬x→0 ⌬x ⌬x

F⬘共x兲 ⫽ lim

⌬x→0





g共x ⫹ ⌬ x兲 ⫺ g共x兲 f 共x ⫹ ⌬ x兲 ⫺ f 共x兲 ⫹ lim g共x兲 ⌬x→0 ⌬x ⌬x g共x ⫹ ⌬ x兲 ⫺ g共x兲 ⫽ lim f 共x ⫹ ⌬ x兲 lim ⌬x→0 ⌬x→0 ⌬x f 共x ⫹ ⌬ x兲 ⫺ f 共x兲 ⫹ lim g共x兲 lim ⌬x→0 ⌬x→0 ⌬x ⫽ f 共x兲g⬘共x兲 ⫹ g共x兲f⬘共x兲 ⫽ lim f 共x ⫹ ⌬ x兲 ⌬x→0



冥冤



冥冤





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154

CHAPTER 2

Differentiation

Example 1

Finding the Derivative of a Product

Find the derivative of y ⫽ 共3x ⫺ 2x2兲共5 ⫹ 4x兲. SOLUTION

Using the Product Rule, you can write First

Derivative of second

Second

Derivative of first

dy d d ⫽ 共3x ⫺ 2x 2兲 关5 ⫹ 4x兴 ⫹ 共5 ⫹ 4x兲 关3x ⫺ 2x 2兴 dx dx dx ⫽ 共3x ⫺ 2x 2兲共4兲 ⫹ 共5 ⫹ 4x兲共3 ⫺ 4x兲 ⫽ 共12x ⫺ 8x 2兲 ⫹ 共15 ⫺ 8x ⫺ 16x 2兲 ⫽ 15 ⫹ 4x ⫺ 24x 2.

✓CHECKPOINT 1 Find the derivative of y ⫽ 共4x ⫹ 3x2兲共6 ⫺ 3x兲.



STUDY TIP In general, the derivative of the product of two functions is not equal to the product of the derivatives of the two functions. To see this, compare the product of the derivatives of f 共x兲 ⫽ 3x ⫺ 2x 2 and g共x兲 ⫽ 5 ⫹ 4x with the derivative found in Example 1.

In the next example, notice that the first step in differentiating is rewriting the original function. TECHNOLOGY If you have access to a symbolic differentiation utility, try using it to confirm several of the derivatives in this section. The form of the derivative can depend on how you use software.

Example 2

Finding the Derivative of a Product

Find the derivative of f 共x兲 ⫽

冢1x ⫹ 1冣共x ⫺ 1兲.

Original function

SOLUTION Rewrite the function. Then use the Product Rule to find the derivative.

f 共x兲 ⫽ 共x⫺1 ⫹ 1兲共x ⫺ 1兲

Rewrite function.

d d 关x ⫺ 1兴 ⫹ 共x ⫺ 1兲 关x⫺1 ⫹ 1兴 dx dx ⫽ 共x⫺1 ⫹ 1兲共1兲 ⫹ 共x ⫺ 1兲共⫺x⫺2兲

f⬘共x兲 ⫽ 共x⫺1 ⫹ 1兲

✓CHECKPOINT 2 Find the derivative of f 共x兲 ⫽

冢1x ⫹ 1冣共2x ⫹ 1兲. ■

Product Rule



1 x⫺1 ⫹1⫺ x x2



x ⫹ x2 ⫺ x ⫹ 1 x2

Write with common denominator.



x2 ⫹ 1 x2

Simplify.

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SECTION 2.4

The Product and Quotient Rules

155

You now have two differentiation rules that deal with products—the Constant Multiple Rule and the Product Rule. The difference between these two rules is that the Constant Multiple Rule deals with the product of a constant and a variable quantity: Variable quantity

Constant

F共x兲 ⫽ c f 共x兲

Use Constant Multiple Rule.

whereas the Product Rule deals with the product of two variable quantities: Variable quantity

Variable quantity

F共x兲 ⫽ f 共x兲 g共x兲.

Use Product Rule.

The next example compares these two rules. STUDY TIP You could calculate the derivatives in Example 3 without the Product Rule. For Example 3(a), y ⫽ 2x共x 2 ⫹ 3x兲 ⫽ 2x3 ⫹ 6x 2 and

Comparing Differentiation Rules

Find the derivative of each function. a. y ⫽ 2x共x 2 ⫹ 3x兲 b. y ⫽ 2共x 2 ⫹ 3x兲 SOLUTION

dy ⫽ 6x 2 ⫹ 12x. dx

a. By the Product Rule, dy d d ⫽ 共2x兲 关x 2 ⫹ 3x兴 ⫹ 共x 2 ⫹ 3x兲 关2x兴 dx dx dx ⫽ 共2x兲共2x ⫹ 3兲 ⫹ 共x 2 ⫹ 3x兲共2兲 ⫽ 4x 2 ⫹ 6x ⫹ 2x 2 ⫹ 6x ⫽ 6x 2 ⫹ 12x.

✓CHECKPOINT 3

Product Rule

b. By the Constant Multiple Rule,

Find the derivative of each function. a. y ⫽ 3x共2x2 ⫹ 5x兲 b. y ⫽ 3共2x2 ⫹ 5x兲

Example 3



dy d ⫽ 2 关x 2 ⫹ 3x兴 dx dx ⫽ 2共2x ⫹ 3兲 ⫽ 4x ⫹ 6.

Constant Multiple Rule

The Product Rule can be extended to products that have more than two factors. For example, if f, g, and h are differentiable functions of x, then d 关 f 共x兲g共x兲h共x兲兴 ⫽ f⬘共x兲g共x兲h共x兲 ⫹ f 共x兲g⬘共x兲h共x兲 ⫹ f 共x兲g共x兲h⬘共x兲. dx

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156

CHAPTER 2

Differentiation

The Quotient Rule In Section 2.2, you saw that by using the Constant Rule, the Power Rule, the Constant Multiple Rule, and the Sum and Difference Rules, you were able to differentiate any polynomial function. By combining these rules with the Quotient Rule, you can now differentiate any rational function. The Quotient Rule

The derivative of the quotient of two differentiable functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. d f 共x兲 g共x兲 f⬘共x兲 ⫺ f 共x兲g⬘共x兲 ⫽ , dx g共x兲 关g共x兲兴2

冤 冥

g共x兲 ⫽ 0

STUDY TIP From this differentiation rule, you can see that the derivative of a quotient is not, in general, the quotient of the derivatives. That is, d f 共x兲 f⬘共x兲 ⫽ . dx g共x兲 g⬘共x兲

冤 冥

Let F共x兲 ⫽ f 共x兲兾g共x兲. As in the proof of the Product Rule, a key step in this proof is adding and subtracting the same quantity. PROOF

F共x ⫹ ⌬x兲 ⫺ F共x兲 ⌬x f 共x ⫹ ⌬x兲 f 共x兲 ⫺ g共x ⫹ ⌬x兲 g共x兲 ⫽ lim ⌬x→0 ⌬x

F⬘共x兲 ⫽ lim

⌬x→0

⫽ lim

g共x兲 f 共x ⫹ ⌬x兲 ⫺ f 共x兲g共x ⫹ ⌬x兲 ⌬xg共x兲g共x ⫹ ⌬x兲

⫽ lim

g共x兲 f 共x ⫹ ⌬x兲 ⫺ f 共x兲g共x兲 ⫹ f 共x兲g共x兲 ⫺ f 共x兲g共x ⫹ ⌬x兲 ⌬xg共x兲g共x ⫹ ⌬x兲

⌬x→0

⌬x→0

lim

STUDY TIP As suggested for the Product Rule, it can be more helpful to remember the verbal statement of the Quotient Rule rather than trying to remember the formula for the rule.



⌬x→0

g共x兲关 f 共x ⫹ ⌬x兲 ⫺ f 共x兲兴 f 共x兲关g共x ⫹ ⌬x兲 ⫺ g共x兲兴 ⫺ lim ⌬x→0 ⌬x ⌬x lim 关g共x兲g共x ⫹ ⌬x兲兴 ⌬x→0





g共x兲 lim

⌬x→0

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 g共x ⫹ ⌬x兲 ⫺ g共x兲 ⫺ f 共x兲 lim ⌬x→0 ⌬x ⌬x lim 关g共x兲g共x ⫹ ⌬x兲兴







⌬x→0



g共x兲 f⬘共x兲 ⫺ f 共x兲g⬘共x兲 关g共x兲兴2

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SECTION 2.4

Example 4

Algebra Review When applying the Quotient Rule, it is suggested that you enclose all factors and derivatives in symbols of grouping, such as parentheses. Also, pay special attention to the subtraction required in the numerator. For help in evaluating expressions like the one in Example 4, see the Chapter 2 Algebra Review on page 197, Example 2(d).

157

Finding the Derivative of a Quotient

Find the derivative of y ⫽

x⫺1 . 2x ⫹ 3

Apply the Quotient Rule, as shown.

SOLUTION

dy ⫽ dx

The Product and Quotient Rules

共2x ⫹ 3兲

d d 关x ⫺ 1兴 ⫺ 共x ⫺ 1兲 关2x ⫹ 3兴 dx dx 共2x ⫹ 3兲2

共2x ⫹ 3兲共1兲 ⫺ 共x ⫺ 1兲共2兲 共2x ⫹ 3兲2 2x ⫹ 3 ⫺ 2x ⫹ 2 ⫽ 共2x ⫹ 3兲2 5 ⫽ 共2x ⫹ 3兲2 ⫽

✓CHECKPOINT 4 Find the derivative of y ⫽

y=

2x 2 − 4x + 3 2 − 3x

Example 5

y

4

−4

−2



Finding an Equation of a Tangent Line

Find an equation of the tangent line to the graph of

6

−6

x⫹4 . 5x ⫺ 2

y⫽ 4

6

x

−2

2x 2 ⫺ 4x ⫹ 3 2 ⫺ 3x

when x ⫽ 1. SOLUTION

Apply the Quotient Rule, as shown.

−4

FIGURE 2.27

dy ⫽ dx

共2 ⫺ 3x兲

d d 关2x 2 ⫺ 4x ⫹ 3兴 ⫺ 共2x 2 ⫺ 4x ⫹ 3兲 关2 ⫺ 3x兴 dx dx 共2 ⫺ 3x兲2

共2 ⫺ 3x兲共4x ⫺ 4兲 ⫺ 共2x 2 ⫺ 4x ⫹ 3兲共⫺3兲 共2 ⫺ 3x兲2 ⫺12x 2 ⫹ 20x ⫺ 8 ⫺ 共⫺6x 2 ⫹ 12x ⫺ 9兲 ⫽ 共2 ⫺ 3x兲2 ⫺12x 2 ⫹ 20x ⫺ 8 ⫹ 6x 2 ⫺ 12x ⫹ 9 ⫽ 共2 ⫺ 3x兲2 ⫺6x 2 ⫹ 8x ⫹ 1 ⫽ 共2 ⫺ 3x兲2 ⫽

✓CHECKPOINT 5 Find an equation of the tangent line to the graph of y⫽

x2 ⫺ 4 when x ⫽ 0. 2x ⫹ 5

Sketch the line tangent to the graph at x ⫽ 0. ■

When x ⫽ 1, the value of the function is y ⫽ ⫺1 and the slope is m ⫽ 3. Using the point-slope form of a line, you can find the equation of the tangent line to be y ⫽ 3x ⫺ 4. The graph of the function and the tangent line is shown in Figure 2.27.

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158

CHAPTER 2

Differentiation

Example 6

Finding the Derivative of a Quotient

Find the derivative of y⫽

3 ⫺ 共1兾x兲 . x⫹5

SOLUTION Begin by rewriting the original function. Then apply the Quotient Rule and simplify the result.

3 ⫺ 共1兾x兲 x⫹5

Write original function.



3x ⫺ 1 x共x ⫹ 5兲

Multiply numerator and denominator by x.



3x ⫺ 1 x 2 ⫹ 5x

Rewrite.

y⫽

dy 共x 2 ⫹ 5x兲共3兲 ⫺ 共3x ⫺ 1兲共2x ⫹ 5兲 ⫽ dx 共x 2 ⫹ 5x兲2

共3x 2 ⫹ 15x兲 ⫺ 共6x 2 ⫹ 13x ⫺ 5兲 共x 2 ⫹ 5x兲2 ⫺3x 2 ⫹ 2x ⫹ 5 ⫽ 共x 2 ⫹ 5x兲2

Apply Quotient Rule.



Simplify.

✓CHECKPOINT 6 Find the derivative of y ⫽

3 ⫺ 共2兾x兲 . x⫹4



Not every quotient needs to be differentiated by the Quotient Rule. For instance, each of the quotients in the next example can be considered as the product of a constant and a function of x. In such cases, the Constant Multiple Rule is more efficient than the Quotient Rule. STUDY TIP To see the efficiency of using the Constant Multiple Rule in Example 7, try using the Quotient Rule to find the derivatives of the four functions.

✓CHECKPOINT 7 Find the derivative of each function. x 2 ⫹ 4x a. y ⫽ 5

3x 4 b. y ⫽ 4

Example 7

Rewriting Before Differentiating

Find the derivative of each function. Original Function x 2 ⫹ 3x a. y ⫽ 6

Rewrite

Differentiate

Simplify

1 y ⫽ 共x 2 ⫹ 3x兲 6

1 y⬘ ⫽ 共2x ⫹ 3兲 6

y⬘ ⫽

5 y⬘ ⫽ 共4x3兲 8

5 y⬘ ⫽ x3 2

1 1 x⫹ 3 2

b. y ⫽

5x 4 8

y⫽

c. y ⫽

⫺3共3x ⫺ 2x 2兲 7x

3 y ⫽ ⫺ 共3 ⫺ 2x兲 7

3 y⬘ ⫽ ⫺ 共⫺2兲 7

y⬘ ⫽

d. y ⫽

9 5x 2

9 y ⫽ 共x⫺2兲 5

9 y⬘ ⫽ 共⫺2x⫺3兲 5

y⬘ ⫽ ⫺

5 4 x 8

6 7 18 5x3



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.4

The Product and Quotient Rules

159

Simplifying Derivatives Example 8

Combining the Product and Quotient Rules

Find the derivative of y⫽

共1 ⫺ 2x兲共3x ⫹ 2兲 . 5x ⫺ 4

This function contains a product within a quotient. You could first multiply the factors in the numerator and then apply the Quotient Rule. However, to gain practice in using the Product Rule within the Quotient Rule, try differentiating as shown. SOLUTION

y⬘ ⫽

共5x ⫺ 4兲

d d 关共1 ⫺ 2x兲共3x ⫹ 2兲兴 ⫺ 共1 ⫺ 2x兲共3x ⫹ 2兲 关5x ⫺ 4兴 dx dx 共5x ⫺ 4兲2

共5x ⫺ 4兲关共1 ⫺ 2x兲共3兲 ⫹ 共3x ⫹ 2兲共⫺2兲兴 ⫺ 共1 ⫺ 2x兲共3x ⫹ 2兲共5兲 共5x ⫺ 4兲2 共5x ⫺ 4兲共⫺12x ⫺ 1兲 ⫺ 共1 ⫺ 2x兲共15x ⫹ 10兲 ⫽ 共5x ⫺ 4兲2 共⫺60x 2 ⫹ 43x ⫹ 4兲 ⫺ 共⫺30x 2 ⫺ 5x ⫹ 10兲 ⫽ 共5x ⫺ 4兲2 ⫺30x 2 ⫹ 48x ⫺ 6 ⫽ 共5x ⫺ 4兲2 ⫽

✓CHECKPOINT 8 Find the derivative of y ⫽

共1 ⫹ x兲共2x ⫺ 1兲 . x⫺1



In the examples in this section, much of the work in obtaining the final form of the derivative occurs after the differentiation. As summarized in the list below, direct application of differentiation rules often yields results that are not in simplified form. Note that two characteristics of simplified form are the absence of negative exponents and the combining of like terms. f⬘共x兲 After Differentiating

f⬘共x兲 After Simplifying

Example 1

共3x ⫺ 2x 2兲共4兲 ⫹ 共5 ⫹ 4x兲共3 ⫺ 4x兲

15 ⫹ 4x ⫺ 24x 2

Example 2

共x⫺1 ⫹ 1兲共1兲 ⫹ 共x ⫺ 1兲共⫺x⫺2兲

x2 ⫹ 1 x2

Example 5

共2 ⫺ 3x兲共4x ⫺ 4兲 ⫺ 共2x 2 ⫺ 4x ⫹ 3兲共⫺3兲 共2 ⫺ 3x兲2

⫺6x 2 ⫹ 8x ⫹ 1 共2 ⫺ 3x兲2

Example 8

共5x ⫺ 4兲关共1 ⫺ 2x兲共3兲 ⫹ 共3x ⫹ 2兲共⫺2兲兴 ⫺ 共1 ⫺ 2x兲共3x ⫹ 2兲共5兲 共5x ⫺ 4兲2

⫺30x 2 ⫹ 48x ⫺ 6 共5x ⫺ 4兲2

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160

CHAPTER 2

Differentiation

Application Example 9

Rate of Change of Systolic Blood Pressure

As blood moves from the heart through the major arteries out to the capillaries and back through the veins, the systolic blood pressure continuously drops. Consider a person whose systolic blood pressure P (in millimeters of mercury) is given by aorta

25t2 ⫹ 125 , 0 ≤ t ≤ 10 t2 ⫹ 1 where t is measured in seconds. At what rate is the blood pressure changing 5 seconds after blood leaves the heart? P⫽

artery vein

SOLUTION

Begin by applying the Quotient Rule.

dP 共t ⫹ 1兲共50t兲 ⫺ 共25t 2 ⫹ 125兲共2t兲 ⫽ dt 共t 2 ⫹ 1兲2 2

artery



vein

Quotient Rule

50t 3 ⫹ 50t ⫺ 50t 3 ⫺ 250t 共t 2 ⫹ 1兲2

200t 共 ⫹ 1兲2 When t ⫽ 5, the rate of change is ⫽⫺

Simplify.

t2

200共5兲 ⬇ ⫺1.48 millimeters per second. 262 So, the pressure is dropping at a rate of 1.48 millimeters per second when t ⫽ 5 seconds. ⫺

✓CHECKPOINT 9 In Example 9, find the rate at which systolic blood pressure is changing at each time shown in the table below. Describe the changes in blood pressure as the blood moves away from the heart. t

0

1

2

3

4

5

dP dt

6

7



CONCEPT CHECK 1. Write a verbal statement that represents the Product Rule. 2. Write a verbal statement that represents the Quotient Rule. x3 1 5x 3. Is it possible to find the derivative of f 冇x冈 ⴝ without using the 2 Quotient Rule? If so, what differentiation rule can you use to find f⬘ ? (You do not need to find the derivative.) 4. Complete the following: In general, you can use the Product Rule to differentiate the ______ of two variable quantities and the Quotient Rule to differentiate any ______ function.

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SECTION 2.4

Skills Review 2.4

161

The Product and Quotient Rules

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 0.4, 0.5, and 2.2.

In Exercises 1–10, simplify the expression. 1. 共x 2 ⫹ 1兲共2兲 ⫹ 共2x ⫹ 7兲共2x兲

2. 共2x ⫺ x3兲共8x兲 ⫹ 共4x 2兲共2 ⫺ 3x 2兲

3. x共4兲共x 2 ⫹ 2兲3共2x兲 ⫹ 共x 2 ⫹ 4兲共1兲

4. x 2共2兲共2x ⫹ 1兲共2兲 ⫹ 共2x ⫹ 1兲4共2x兲

5.

共2x ⫹ 7兲共5兲 ⫺ 共5x ⫹ 6兲共2兲 共2x ⫹ 7兲2

6.

共x 2 ⫺ 4兲共2x ⫹ 1兲 ⫺ 共x 2 ⫹ x兲共2x兲 共x 2 ⫺ 4兲2

7.

共x 2 ⫹ 1兲共2兲 ⫺ 共2x ⫹ 1兲共2x兲 共x 2 ⫹ 1兲2

8.

共1 ⫺ x 4兲共4兲 ⫺ 共4x ⫺ 1兲共⫺4x 3兲 共1 ⫺ x 4兲2

9. 共x⫺1 ⫹ x兲共2兲 ⫹ 共2x ⫺ 3兲共⫺x⫺2 ⫹ 1兲

10.

共1 ⫺ x⫺1兲共1兲 ⫺ 共x ⫺ 4兲共x⫺2兲 共1 ⫺ x⫺1兲 2

In Exercises 11–14, find f⬘ 冇2冈. 11. f 共x兲 ⫽ 3x 2 ⫺ x ⫹ 4 13. f 共x兲 ⫽

12. f 共x兲 ⫽ ⫺x3 ⫹ x 2 ⫹ 8x

1 x

14. f 共x兲 ⫽ x 2 ⫺

Exercises 2.4

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–16, find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. Function

Point

1. f (x) ⫽ x共x ⫹ 3兲 2

1 x2

共2, 14)

2. g共x兲 ⫽ 共x ⫺ 4兲共x ⫹ 2兲

共4, 0兲

3. f 共x兲 ⫽ x 2共3x3 ⫺ 1兲

共1, 2兲

4. f 共x兲 ⫽ 共x 2 ⫹ 1兲共2x ⫹ 5兲

共⫺1, 6兲

Function

Point

t2 ⫺ 1 15. f 共t兲 ⫽ t⫹4

共1, 0兲

16. g共x兲 ⫽

4x ⫺ 5 x2 ⫺ 1

共0, 5兲

In Exercises 17–24, find the derivative of the function. Use Example 7 as a model. Function

Rewrite

Differentiate

Simplify

共0, ⫺ 43 兲 共1, ⫺ 17 兲

17. y ⫽

x ⫹ 2x x

䊏 䊏



7. g共x兲 ⫽ 共x 2 ⫺ 4x ⫹ 3兲共x ⫺ 2兲

共4, 6兲

8. g共x兲 ⫽ 共x 2 ⫺ 2x ⫹ 1兲共x3 ⫺ 1兲

共1, 0兲

18. y ⫽

4x3兾2 x

䊏 䊏



19. y ⫽

7 3x3

䊏 䊏



冢⫺1, 21冣 冢3, 32冣 冢⫺1, ⫺ 35冣

20. y ⫽

4 5x 2

䊏 䊏



䊏 䊏



䊏 䊏



共6, 13兲

23. y ⫽

x 2 ⫺ 4x ⫹ 3 x⫺1

䊏 䊏



共2, 3兲

24. y ⫽

x2 ⫺ 4 x⫹2

䊏 䊏



5. f 共x兲 ⫽ 6. f 共x兲 ⫽

x x⫺5 x2 h共x兲 ⫽ x⫹3 2t 2 ⫺ 3 f 共t兲 ⫽ 3t ⫹ 1 3x f 共x兲 ⫽ 2 x ⫹4 2x ⫹ 1 g共x兲 ⫽ x⫺5 x⫹1 f 共x兲 ⫽ x⫺1

9. h共x兲 ⫽ 10. 11. 12. 13. 14.

1 3 3 共2x ⫺ 4兲 1 2 7 共5 ⫺ 6x 兲

共6, 6兲

2

4x 2 ⫺ 3x 8冪x 3x 2 ⫺ 4x 22. y ⫽ 6x 21. y ⫽

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162

CHAPTER 2

Differentiation

In Exercises 25– 40, find the derivative of the function. State which differentiation rule(s) you used to find the derivative. 25. f 共x兲 ⫽ 共x3 ⫺ 3x兲共2x 2 ⫹ 3x ⫹ 5兲 26. h共t兲 ⫽ 共 ⫺ 1兲共 t5

4t2

3 x 冪x ⫹ 3 29. f 共x兲 ⫽ 冪 共 兲

31. f 共x兲 ⫽

3x ⫺ 2 2x ⫺ 3

28. h共 p兲 ⫽ 共 p3 ⫺ 2兲2

3 ⫺ 2x ⫺ x 2 33. f 共x兲 ⫽ x2 ⫺ 1



37. g共s兲 ⫽

s 2 ⫺ 2s ⫹ 5 冪s

39. g共x兲 ⫽

冢xx ⫺⫹ 34冣 共x

x3 ⫹ 3x ⫹ 2 x2 ⫺ 1

冢 冣

1 34. f 共x兲 ⫽ 共x5 ⫺ 3x兲 2 x

2 35. f 共x兲 ⫽ x 1 ⫺ x⫹1



t⫹2 36. h共t兲 ⫽ 2 t ⫹ 5t ⫹ 6 38. f 共x兲 ⫽

x⫹1 冪x

⫹ 2x ⫹ 1兲

In Exercises 41– 46, find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. Point

41. f 共x兲 ⫽ 共x ⫺ 1兲 共x ⫺ 2兲

共0, ⫺2兲

42. h共x兲 ⫽ 共x 2 ⫺ 1兲2

共⫺2, 9兲

2

56. x ⫽ 300 ⫺ p ⫺

2p , p ⫽ $3 p⫹1

57. Environment

The model

f 共t兲 ⫽

t2

⫺t⫹1 t2 ⫹ 1

measures the level of oxygen in a pond, where t is the time (in weeks) after organic waste is dumped into the pond. Find the rates of change of f with respect to t when (a) t ⫽ 0.5, (b) t ⫽ 2, and (c) t ⫽ 8. 58. Physical Science The temperature T (in degrees Fahrenheit) of food placed in a refrigerator is modeled by T ⫽ 10

40. f 共x兲 ⫽ 共3x3 ⫹ 4x兲共x ⫺ 5兲共x ⫹ 1兲

Function



3p , p ⫽ $4 5p ⫹ 1

3 x 共x ⫹ 1兲 30. f 共x兲 ⫽ 冪

32. f 共x兲 ⫽

2



55. x ⫽ 275 1 ⫺

⫺ 7t ⫺ 3兲

27. g共t兲 ⫽ 共2t 3 ⫺ 1兲2

Demand In Exercises 55 and 56, use the demand function to find the rate of change in the demand x for the given price p.

2

冢4tt

2

⫹ 16t ⫹ 75 ⫹ 4t ⫹ 10



where t is the time (in hours). What is the initial temperature of the food? Find the rates of change of T with respect to t when (a) t ⫽ 1, (b) t ⫽ 3, (c) t ⫽ 5, and (d) t ⫽ 10. 59. Population Growth A population of bacteria is introduced into a culture. The number of bacteria P can be modeled by



P ⫽ 500 1 ⫹

4t 50 ⫹ t 2



where t is the time (in hours). Find the rate of change of the population when t ⫽ 2.

43. f 共x兲 ⫽

x⫺2 x⫹1

共1, ⫺ 12 兲

44. f 共x兲 ⫽

2x ⫹ 1 x⫺1

共2, 5兲

45. f 共x兲 ⫽

冢xx ⫹⫺ 51冣共2x ⫹ 1兲

共0, ⫺5兲

P⫽

共0, ⫺10兲

Find the rates of change of P when (a) t ⫽ 1 and (b) t ⫽ 10.

46. g共x兲 ⫽ 共x ⫹ 2兲

冢xx ⫺⫹ 51冣

60. Quality Control The percent P of defective parts produced by a new employee t days after the employee starts work can be modeled by

In Exercises 47–50, find the point(s), if any, at which the graph of f has a horizontal tangent. 47. f 共x兲 ⫽ 49. f 共x兲 ⫽

x2 x⫺1 x3

x4 ⫹1

48. f 共x兲 ⫽ 50. f 共x兲 ⫽

x2

x2 ⫹1

x4 ⫹ 3 x2 ⫹ 1

In Exercises 51–54, use a graphing utility to graph f and f⬘ on the interval [ⴚ2, 2]. 51. f 共x兲 ⫽ x共x ⫹ 1兲

52. f 共x兲 ⫽ x 2共x ⫹ 1兲

53. f 共x兲 ⫽ x共x ⫹ 1兲共x ⫺ 1兲

54. f 共x兲 ⫽ x 2共x ⫹ 1兲共x ⫺ 1兲

t ⫹ 1750 . 50共t ⫹ 2兲

61. MAKE A DECISION: NEGOTIATING A PRICE You decide to form a partnership with another business. Your business determines that the demand x for your product is inversely proportional to the square of the price for x ≥ 5. (a) The price is $1000 and the demand is 16 units. Find the demand function. (b) Your partner determines that the product costs $250 per unit and the fixed cost is $10,000. Find the cost function. (c) Find the profit function and use a graphing utility to graph it. From the graph, what price would you negotiate with your partner for this product? Explain your reasoning.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.4 62. Managing a Store You are managing a store and have been adjusting the price of an item. You have found that you make a profit of $50 when 10 units are sold, $60 when 12 units are sold, and $65 when 14 units are sold. (a) Fit these data to the model P ⫽ ax 2 ⫹ bx ⫹ c. (b) Use a graphing utility to graph P. (c) Find the point on the graph at which the marginal profit is zero. Interpret this point in the context of the problem. 63. Demand Function Given f 共x兲 ⫽ x ⫹ 1, which function would most likely represent a demand function? Explain your reasoning. Use a graphing utility to graph each function, and use each graph as part of your explanation. (a) p ⫽ f 共x兲

(b) p ⫽ x f 共x兲

(c) p ⫽ ⫺f 共x兲 ⫹ 5

64. Cost The cost of producing x units of a product is given by

The Product and Quotient Rules

68. Sales Analysis The monthly sales of memberships M at a newly built fitness center are modeled by M共t兲 ⫽

300t ⫹8 t2 ⫹ 1

where t is the number of months since the center opened. (a) Find M⬘共t兲. (b) Find M共3兲 and M⬘共3兲 and interpret the results. (c) Find M共24兲 and M⬘共24兲 and interpret the results. In Exercises 69–72, use the given information to find f⬘冇2冈. g冇2冈 ⴝ 3 h冇2冈 ⴝ ⴚ1

and and

g⬘冇2冈 ⴝ ⴚ2 h⬘冇2冈 ⴝ 4

C ⫽ x ⫺ 15x ⫹ 87x ⫺ 73, 4 ≤ x ≤ 9.

69. f 共x兲 ⫽ 2g共x) ⫹ h共x)

70. f 共x) ⫽ 3 ⫺ g共x)

(a) Use a graphing utility to graph the marginal cost function and the average cost function, C兾x, in the same viewing window.

71. f (x兲 ⫽ g(x) ⫹ h(x兲

72. f 共x兲 ⫽

3

2

(b) Find the point of intersection of the graphs of dC兾dx and C兾x. Does this point have any significance?

163

g共x兲 h共x兲

Business Capsule

65. MAKE A DECISION: INVENTORY REPLENISHMENT The ordering and transportation cost C per unit (in thousands of dollars) of the components used in manufacturing a product is given by C ⫽ 100

x ⫹ , 冢200 x x ⫹ 30 冣 2

1 ≤ x

where x is the order size (in hundreds). Find the rate of change of C with respect to x for each order size. What do these rates of change imply about increasing the size of an order? Of the given order sizes, which would you choose? Explain. (a) x ⫽ 10

(b) x ⫽ 15

(c) x ⫽ 20

66. Inventory Replenishment The ordering and transportation cost C per unit for the components used in manufacturing a product is C ⫽ 共375,000 ⫹ 6x 2兲兾x,

x ≥ 1

where C is measured in dollars and x is the order size. Find the rate of change of C with respect to x when (a) x ⫽ 200, (b) x ⫽ 250, and (c) x ⫽ 300. Interpret the meaning of these values. 67. Consumer Awareness The prices per pound of lean and extra lean ground beef in the United States from 1998 to 2005 can be modeled by P⫽

1.755 ⫺ 0.2079t ⫹ 0.00673t2 , 1 ⫺ 0.1282t ⫹ 0.00434t 2

8 ≤ t ≤ 15

where t is the year, with t ⫽ 8 corresponding to 1998. Find dP兾dt and evaluate it for t ⫽ 8, 10, 12, and 14. Interpret the meaning of these values. (Source: U.S. Bureau of Labor Statistics)

AP/Wide World Photos

n 1978 Ben Cohen and Jerry Greenfield used their combined life savings of $8000 to convert an abandoned gas station in Burlington, Vermont into their first ice cream shop. Today, Ben & Jerry’s Homemade Holdings, Inc. has over 600 scoop shops in 16 countries. The company’s three-part mission statement emphasizes product quality, economic reward, and a commitment to the community. Ben & Jerry’s contributes a minimum of $1.1 million annually through corporate philanthropy that is primarily employee led.

I

73. Research Project Use your school’s library, the Internet, or some other reference source to find information on a company that is noted for its philanthropy and community commitment. (One such business is described above.) Write a short paper about the company.

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164

CHAPTER 2

Differentiation

Mid-Chapter Quiz

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Take this quiz as you would take a quiz in class. When you are done, check your work against the answers given in the back of the book. In Exercises 1–3, use the limit definition to find the derivative of the function. Then find the slope of the tangent line to the graph of f at the given point. 1. f 共x兲 ⫽ ⫺x ⫹ 2; 共2, 0兲

2. f 共x兲 ⫽ 冪x ⫹ 3; 共1, 2)

4 3. f 共x兲 ⫽ ; 共1, 4) x

In Exercises 4 –12, find the derivative of the function. 4. f (x) ⫽ 12 7. f (x) ⫽ 12x 10. f 共x兲 ⫽

1兾4

2x ⫹ 3 3x ⫹ 2

5. f 共x) ⫽ 19x ⫹ 9

6. f 共x兲 ⫽ 5 ⫺ 3x2

8. f (x) ⫽ 4x

9. f (x) ⫽ 2冪x

⫺2

11. f (x兲 ⫽ 共x2 ⫹ 1兲共⫺2x ⫹ 4)

12. f 共x兲 ⫽

4⫺x x⫹5

In Exercises 13–16, use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. 13. f 共x兲 ⫽ x2 ⫺ 3x ⫹ 1; 关0, 3兴 14. f 共x兲 ⫽ 2x3 ⫹ x2 ⫺ x ⫹ 4; 关⫺1, 1兴 15. f 共x兲 ⫽

1 ; [2, 5兴 2x

3 x; 关8, 27兴 16. f 共x兲 ⫽ 冪

17. The profit (in dollars) from selling x units of a product is given by P ⫽ ⫺0.0125x2 ⫹ 16x ⫺ 600 (a) Find the additional profit when the sales increase from 175 to 176 units. (b) Find the marginal profit when x ⫽ 175. (c) Compare the results of parts (a) and (b). In Exercises 18 and 19, find an equation of the tangent line to the graph of f at the given point. Then use a graphing utility to graph the function and the equation of the tangent line in the same viewing window. 18. f 共x) ⫽ 5x2 ⫹ 6x ⫺ 1; 共⫺1, ⫺2兲 19. f (x兲 ⫽ 共x ⫺ 1兲共x ⫹ 1); 共0, ⫺1兲 20. From 2000 through 2005, the sales per share S (in dollars) for CVS Corporation can be modeled by S ⫽ 0.18390t 3 ⫺ 0.8242t2 ⫹ 3.492t ⫹ 25.60, 0 ≤ t ≤ 5 where t represents the year, with t ⫽ 0 corresponding to 2000. Corporation)

(Source: CVS

(a) Find the rate of change of the sales per share with respect to the year. (b) At what rate were the sales per share changing in 2001? in 2004? in 2005?

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SECTION 2.5

The Chain Rule

165

Section 2.5

The Chain Rule

■ Find derivatives using the Chain Rule. ■ Find derivatives using the General Power Rule. ■ Write derivatives in simplified form. ■ Use derivatives to answer questions about real-life situations. ■ Use the differentiation rules to differentiate algebraic functions.

The Chain Rule In this section, you will study one of the most powerful rules of differential calculus—the Chain Rule. This differentiation rule deals with composite functions and adds versatility to the rules presented in Sections 2.2 and 2.4. For example, compare the functions below. Those on the left can be differentiated without the Chain Rule, whereas those on the right are best done with the Chain Rule.

x Input Function g

Rate of change of u with respect to x is du . dx

Without the Chain Rule

With the Chain Rule

y ⫽ x2 ⫹ 1

y ⫽ 冪x2 ⫹ 1

y⫽x⫹1 y ⫽ 3x ⫹ 2

y ⫽ 共x ⫹ 1兲⫺1兾2 y ⫽ 共3x ⫹ 2兲5 x⫹5 2 y⫽ 2 x ⫹2

y⫽

x⫹5 x2 ⫹ 2





The Chain Rule

If y ⫽ f 共u兲 is a differentiable function of u, and u ⫽ g共x兲 is a differentiable function of x, then y ⫽ f 共g共x兲兲 is a differentiable function of x, and

Output u = g(x) u Input Function f

Rate of change of y with respect to u is dy . du

dy dy ⫽ dx du



du dx

or, equivalently, d 关 f 共g共x兲兲兴 ⫽ f⬘共g共x兲兲g⬘共x兲. dx Basically, the Chain Rule states that if y changes dy兾du times as fast as u, and u changes du兾dx times as fast as x, then y changes

Output Rate of change of y with respect to x is dy dy du = . dx du dx

FIGURE 2.28

y = f (u) = f (g (x))

dy du



du dx

times as fast as x, as illustrated in Figure 2.28. One advantage of the dy兾dx notation for derivatives is that it helps you remember differentiation rules, such as the Chain Rule. For instance, in the formula dy兾dx ⫽ 共dy兾du兲共du兾dx兲 you can imagine that the du’s divide out.

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166

CHAPTER 2

Differentiation

When applying the Chain Rule, it helps to think of the composite function y ⫽ f 共g共x兲兲 or y ⫽ f 共u兲 as having two parts—an inside and an outside—as illustrated below. Inside

y ⫽ f 共g共x兲兲 ⫽ f 共u兲 Outside

The Chain Rule tells you that the derivative of y ⫽ f 共u兲 is the derivative of the outer function (at the inner function u) times the derivative of the inner function. That is, y⬘ ⫽ f⬘共u兲 ⭈ u⬘.

✓CHECKPOINT 1

Example 1

Write each function as the composition of two functions, where y ⫽ f 共g共x兲兲. 1 a. y ⫽ 冪x ⫹ 1 b. y ⫽ 共

x2

Decomposing Composite Functions

Write each function as the composition of two functions. a. y ⫽

1 x⫹1

b. y ⫽ 冪3x2 ⫺ x ⫹ 1

There is more than one correct way to decompose each function. One way for each is shown below. SOLUTION

⫹ 2x ⫹ 5兲

3



y ⫽ f 共g共x兲兲 a. y ⫽

1 x⫹1

b. y ⫽ 冪3x2 ⫺ x ⫹ 1

Example 2 STUDY TIP Try checking the result of Example 2 by expanding the function to obtain y ⫽ x 6 ⫹ 3x 4 ⫹ 3x2 ⫹ 1 and finding the derivative. Do you obtain the same answer?

u ⫽ g共x兲 (inside)

y ⫽ f 共u兲 (outside)

u⫽x⫹1

y⫽

u ⫽ 3x2 ⫺ x ⫹ 1

y ⫽ 冪u

1 u

Using the Chain Rule

Find the derivative of y ⫽ 共x2 ⫹ 1兲3. SOLUTION

To apply the Chain Rule, you need to identify the inside function u. u

y ⫽ 共x 2 ⫹ 1兲3 ⫽ u3 By the Chain Rule, you can write the derivative as shown. dy du

du dx

dy ⫽ 3共x 2 ⫹ 1兲2共2x兲 ⫽ 6x共x2 ⫹ 1兲2 dx

✓CHECKPOINT 2 Find the derivative of y ⫽ 共x3 ⫹ 1兲2.



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SECTION 2.5

The Chain Rule

167

The General Power Rule The function in Example 2 illustrates one of the most common types of composite functions—a power function of the form y ⫽ 关u共x兲兴 n. The rule for differentiating such functions is called the General Power Rule, and it is a special case of the Chain Rule. The General Power Rule

If y ⫽ 关u共x兲兴n, where u is a differentiable function of x and n is a real number, then dy du ⫽ n关u共x兲兴n⫺1 dx dx or, equivalently, d n 关u 兴 ⫽ nun⫺1u⬘. dx

PROOF

Apply the Chain Rule and the Simple Power Rule as shown.

dy dy ⫽ dx du



du dx

d n du 关u 兴 du dx du ⫽ nun⫺1 dx



TECHNOLOGY If you have access to a symbolic differentiation utility, try using it to confirm the result of Example 3.

Example 3

Using the General Power Rule

Find the derivative of f 共x兲 ⫽ 共3x ⫺ 2x2兲3. SOLUTION The inside function is u ⫽ 3x ⫺ 2x2. So, by the General Power Rule, n

un⫺1

u⬘

d 关3x ⫺ 2x2兴 dx ⫽ 3共3x ⫺ 2x2兲2共3 ⫺ 4x兲 ⫽ 共9 ⫺ 12x兲共3x ⫺ 2x2兲2.

f⬘共x兲 ⫽ 3共3x ⫺ 2x2兲2

✓CHECKPOINT 3 Find the derivative of y ⫽ 共x2 ⫹ 3x兲4.



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168

CHAPTER 2

Differentiation

Example 4

Rewriting Before Differentiating

Find the tangent line to the graph of 3 共x2 ⫹ 4兲2 y⫽冪

Original function

when x ⫽ 2. Begin by rewriting the function in rational exponent form.

SOLUTION

y⫽共

x2

⫹ 4兲2兾3

Rewrite original function.

Then, using the inside function, u ⫽ x2 ⫹ 4, apply the General Power Rule. y=

y

3

(x 2 + 4) 2

n

u⬘

dy 2 2 ⫽ 共x ⫹ 4兲⫺1兾3共2x兲 dx 3 4x共x2 ⫹ 4兲⫺1兾3 ⫽ 3 4x ⫽ 3 冪 3 x2 ⫹ 4

9 8 7 6 5 4 2

− 5 − 4 −3

un⫺1

1 2 3 4 5

x

Apply General Power Rule.

Simplify.

When x ⫽ 2, y ⫽ 4 and the slope of the line tangent to the graph at 共2, 4兲 is 43. Using the point-slope form, you can find the equation of the tangent line to be y ⫽ 43x ⫹ 43. The graph of the function and the tangent line is shown in Figure 2.29.

FIGURE 2.29

✓CHECKPOINT 4 3 共x ⫹ 4兲2 when x ⫽ 4. Sketch the Find the tangent line to the graph of y ⫽ 冪 line tangent to the graph at x ⫽ 4. ■

STUDY TIP The derivative of a quotient can sometimes be found more easily with the General Power Rule than with the Quotient Rule. This is especially true when the numerator is a constant, as shown in Example 5.

Example 5

Finding the Derivative of a Quotient

Find the derivative of each function. a. y ⫽

x2

3 ⫹1

b. y ⫽

3 共x ⫹ 1兲2

SOLUTION

a. Begin by rewriting the function as y ⫽ 3共x2 ⫹ 1兲⫺1.

Rewrite original function.

Then apply the General Power Rule to obtain

✓CHECKPOINT 5 Find the derivative of each function. 4 2x ⫹ 1 2 b. y ⫽ 共x ⫺ 1兲3

dy 6x . ⫽ ⫺3共x2 ⫹ 1兲⫺2共2x兲 ⫽ ⫺ 2 dx 共x ⫹ 1兲2

Apply General Power Rule.

b. Begin by rewriting the function as y ⫽ 3共x ⫹ 1兲⫺2.

Rewrite original function.

Then apply the General Power Rule to obtain

a. y ⫽



dy 6 . ⫽ ⫺6共x ⫹ 1兲⫺3共1兲 ⫽ ⫺ dx 共x ⫹ 1兲3

Apply General Power Rule.

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SECTION 2.5

The Chain Rule

169

Simplification Techniques Throughout this chapter, writing derivatives in simplified form has been emphasized. The reason for this is that most applications of derivatives require a simplified form. The next two examples illustrate some useful simplification techniques.

Algebra Review In Example 6, note that you subtract exponents when factoring. That is, when 共1 ⫺ x2兲⫺1兾2 is factored out of 共1 ⫺ x2兲1兾2, the remaining factor has an exponent of 1 2

⫺ 共⫺ 12 兲 ⫽ 1. So,

共1 ⫺ x2兲1兾2 ⫽ 共1 ⫺ x 2兲⫺1兾2 共1 ⫺ x2兲1. For help in evaluating expressions like the one in Example 6, see the Chapter 2 Algebra Review on pages 196 and 197.

Example 6

Simplifying by Factoring Out Least Powers

Find the derivative of y ⫽ x2冪1 ⫺ x2. y ⫽ x2冪1 ⫺ x2 ⫽ x2共1 ⫺ x2兲1兾2 d d y⬘ ⫽ x2 关共1 ⫺ x2兲1兾2兴 ⫹ 共1 ⫺ x2兲1兾2 关x2兴 dx dx 1 ⫽ x2 共1 ⫺ x2兲⫺1兾2共⫺2x兲 ⫹ 共1 ⫺ x2兲1兾2共2x兲 2 3 ⫽ ⫺x 共1 ⫺ x2兲⫺1兾2 ⫹ 2x共1 ⫺ x2兲1兾2 ⫽ x共1 ⫺ x2兲⫺1兾2关⫺x2共1兲 ⫹ 2共1 ⫺ x2兲兴 ⫽ x共1 ⫺ x2兲⫺1兾2共2 ⫺ 3x2兲 x共2 ⫺ 3x2兲 ⫽ 冪1 ⫺ x 2





Write original function. Rewrite function. Product Rule Power Rule

Factor.

Simplify.

✓CHECKPOINT 6 Find and simplify the derivative of y ⫽ x2冪x2 ⫹ 1. STUDY TIP In Example 7, try to find f⬘共x兲 by applying the Quotient Rule to f 共x兲 ⫽

共3x ⫺ 1兲2 . 共x2 ⫹ 3兲2

Example 7



Differentiating a Quotient Raised to a Power

Find the derivative of f 共x兲 ⫽



3x ⫺ 1 2 . x2 ⫹ 3



SOLUTION

Which method do you prefer?

un⫺1

n

u⬘

冢3xx ⫹⫺ 31冣 dxd 冤 3xx ⫹⫺ 31冥 2共3x ⫺ 1兲 共x ⫹ 3兲共3兲 ⫺ 共3x ⫺ 1兲共2x兲 ⫽冤 冥 x ⫹ 3 冥冤 共x ⫹ 3兲

f⬘共x兲 ⫽ 2

2

2

2

2

✓CHECKPOINT 7 Find the derivative of f 共x兲 ⫽

冢xx ⫹⫺ 15冣 . ■ 2

2

2共3x ⫺ 1兲共

2

⫹9⫺ ⫹ 2x兲 共x2 ⫹ 3兲3 2共3x ⫺ 1兲共⫺3x2 ⫹ 2x ⫹ 9兲 ⫽ 共x2 ⫹ 3兲3 ⫽

3x2

6x2

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CHAPTER 2

Differentiation

Example 8

Finding Rates of Change

From 1996 through 2005, the revenue per share R (in dollars) for U.S. Cellular can be modeled by R ⫽ 共⫺0.009t2 ⫹ 0.54t ⫺ 0.1兲2 for 6 ≤ t ≤ 15, where t is the year, with t ⫽ 6 corresponding to 1996. Use the model to approximate the rates of change in the revenue per share in 1997, 1999, and 2003. If you had been a U.S. Cellular stockholder from 1996 through 2005, would you have been satisfied with the performance of this stock? (Source: U.S. Cellular) The rate of change in R is given by the derivative dR兾dt. You can use the General Power Rule to find the derivative. SOLUTION

dR ⫽ 2共⫺0.009t2 ⫹ 0.54t ⫺ 0.1兲1共⫺0.018t ⫹ 0.54兲 dt ⫽ 共⫺0.036t ⫹ 1.08兲共⫺0.009t2 ⫹ 0.54t ⫺ 0.1兲 In 1997, the revenue per share was changing at a rate of

关⫺0.036共7兲 ⫹ 1.08兴关⫺0.009共7兲2 ⫹ 0.54共7兲 ⫺ 0.1兴 ⬇ $2.68 per year. In 1999, the revenue per share was changing at a rate of

关⫺0.036共9兲 ⫹ 1.08兴关⫺0.009共9兲2 ⫹ 0.54共9兲 ⫺ 0.1兴 ⬇ $3.05 per year. In 2003, the revenue per share was changing at a rate of

关⫺0.036共13兲 ⫹ 1.08兴关⫺0.009共13兲2 ⫹ 0.54共13兲 ⫺ 0.1兴 ⬇ $3.30 per year. The graph of the revenue per share function is shown in Figure 2.30. For most investors, the performance of U.S. Cellular stock would be considered to be good. U.S. Cellular

Revenue per share (in dollars)

170

R 35 30 25 20 15 10 5 6

7

8

9

10

11

12

13

14

15

t

Year (6 ↔ 1996)

FIGURE 2.30

✓CHECKPOINT 8 From 1996 through 2005, the sales per share (in dollars) for Dollar Tree can be modeled by S ⫽ 共⫺0.002t2 ⫹ 0.39t ⫹ 0.1兲2 for 6 ≤ t ≤ 15, where t is the year, with t ⫽ 6 corresponding to 1996. Use the model to approximate the rate of change in sales per share in 2003. (Source: Dollar Tree Stores, Inc.) ■

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SECTION 2.5

The Chain Rule

171

Summary of Differentiation Rules You now have all the rules you need to differentiate any algebraic function. For your convenience, they are summarized below. Summary of Differentiation Rules

Let u and v be differentiable functions of x. 1. Constant Rule

d 关c兴 ⫽ 0, dx

2. Constant Multiple Rule

d du 关cu兴 ⫽ c , c is a constant. dx dx

3. Sum and Difference Rules

d du dv 关u ± v兴 ⫽ ± dx dx dx

4. Product Rule

d dv du 关uv兴 ⫽ u ⫹ v dx dx dx

5. Quotient Rule

d u ⫽ dx v

6. Power Rules

7. Chain Rule

冤冥

v

c is a constant.

du dv ⫺u dx dx v2

d n 关x 兴 ⫽ nx n⫺1 dx d n du 关u 兴 ⫽ nun⫺1 dx dx dy dy ⫽ dx du



du dx

CONCEPT CHECK 1. Write a verbal statement that represents the Chain Rule. 2. Write a verbal statement that represents the General Power Rule. 3. Complete the following: When the numerator of a quotient is a constant, you may be able to find the derivative of the quotient more easily with the ______ ______ Rule than with the Quotient Rule. 4. In the expression f 冇 g冇x冈冈, f is the outer function and g is the inner function. Write a verbal statement of the Chain Rule using the words “inner” and “outer.”

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172

CHAPTER 2

Differentiation The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 0.3 and 0.4.

Skills Review 2.5

In Exercises 1– 6, rewrite the expression with rational exponents. 5 共1 ⫺ 5x兲2 1. 冪

4.

4 共2x ⫺ 1兲3 2. 冪

1

5.

3 冪 x⫺6

3.

冪x

6.

3 冪 1 ⫺ 2x

1 冪4x2 ⫹ 1 冪共3 ⫺ 7x兲3

2x

In Exercises 7–10, factor the expression. 7. 3x3 ⫺ 6x2 ⫹ 5x ⫺ 10 9. 4共

x2

⫹ 1兲 ⫺ x共 2

x2

⫹ 1兲

8. 5x冪x ⫺ x ⫺ 5冪x ⫹ 1 10. ⫺x5 ⫹ 3x3 ⫹ x2 ⫺ 3

3

Exercises 2.5

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 8, identify the inside function, u ⴝ g冇x冈, and the outside function, y ⴝ f 冇u冈. y ⫽ f 共g共x兲兲 1. y ⫽ 共6x ⫺ 5兲4 2. y ⫽ 共x2 ⫺ 2x ⫹ 3兲3 3. y ⫽ 共4 ⫺ x2兲⫺1 4. y ⫽ 共x2 ⫹ 1兲4兾3 5. y ⫽ 冪5x ⫺ 2 6. y ⫽ 冪1 ⫺ x2 7. y ⫽ 共3x ⫹ 1兲⫺1 8. y ⫽ 共x ⫹ 2兲⫺1兾2

u ⫽ g共x兲

y ⫽ f 共u兲

23. y ⫽ 共2x ⫺ 7兲3

䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

䊏 䊏 䊏 䊏 䊏 䊏 䊏 䊏

25. g共x兲 ⫽ 共4 ⫺ 2x兲3

26. h共t兲 ⫽ 共1 ⫺ t 2兲 4

3 2

27. h共x兲 ⫽ 共6x ⫺ x 兲

28. f 共x兲 ⫽ 共4x ⫺ x2兲3

29. f 共x兲 ⫽ 共x2 ⫺ 9兲2兾3

30. f 共t兲 ⫽ 共9t ⫹ 2兲2兾3

31. f 共t兲 ⫽ 冪t ⫹ 1

32. g共x兲 ⫽ 冪5 ⫺ 3x

33. s共t兲 ⫽ 冪2t 2 ⫹ 5t ⫹ 2

3 3x3 ⫹ 4x 34. y ⫽ 冪

35. y ⫽

36. y ⫽ 2冪4 ⫺ x2

In Exercises 9–14, find dy/du, du/dx, and dy/dx. 9. y ⫽ u , u ⫽ 4x ⫹ 7 2

In Exercises 23– 40, use the General Power Rule to find the derivative of the function.

10. y ⫽ u , u ⫽ 3x ⫺ 2 3

2

11. y ⫽ 冪u, u ⫽ 3 ⫺ x2

12. y ⫽ 2冪u, u ⫽ 5x ⫹ 9

13. y ⫽ u2兾3, u ⫽ 5x4 ⫺ 2x

14. y ⫽ u⫺1, u ⫽ x3 ⫹ 2x2

In Exercises 15–22, match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule

(b) Constant Rule

(c) General Power Rule

(d) Quotient Rule

2 15. f 共x兲 ⫽ 1 ⫺ x3

2x 16. f 共x兲 ⫽ 1 ⫺ x3

3 2 8 17. f 共x兲 ⫽ 冪

3 2 x 18. f 共x兲 ⫽ 冪

19. f 共x兲 ⫽

x2 ⫹ 2 x

20. f 共x兲 ⫽

x 4 ⫺ 2x ⫹ 1 冪x

21. f 共x兲 ⫽

2 x⫺2

22. f 共x兲 ⫽

5 x2 ⫹ 1

24. y ⫽ 共2x3 ⫹ 1兲2

9x ⫹ 4

3 冪

2

4 2 ⫺ 9x 37. f 共x兲 ⫽ ⫺3冪

38. f 共x兲 ⫽ 共25 ⫹ x2兲⫺1兾2

39. h共x兲 ⫽ 共4 ⫺ x3兲⫺4兾3

40. f 共x兲 ⫽ 共4 ⫺ 3x兲⫺5兾2

In Exercises 41–46, find an equation of the tangent line to the graph of f at the point 冇2, f 冇2冈冈. Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. 41. f 共x兲 ⫽ 2共x2 ⫺ 1兲3

42. f 共x兲 ⫽ 3共9x ⫺ 4兲4

43. f 共x兲 ⫽ 冪4x2 ⫺ 7

44. f 共x兲 ⫽ x冪x2 ⫹ 5

45. f 共x兲 ⫽ 冪x ⫺ 2x ⫹ 1

46. f 共x兲 ⫽ 共4 ⫺ 3x2兲⫺2兾3

2

In Exercises 47–50, use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. 47. f 共x兲 ⫽ 49. f 共x兲 ⫽

冪x ⫹ 1

x2 ⫹ 1

冪x ⫹x 1

48. f 共x兲 ⫽

冪x 2x⫹ 1

50. f 共x兲 ⫽ 冪x 共2 ⫺ x2兲

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SECTION 2.5 In Exercises 51–66, find the derivative of the function. State which differentiation rule(s) you used to find the derivative. 1 51. y ⫽ x⫺2

1 52. s共t兲 ⫽ 2 t ⫹ 3t ⫺ 1

4 53. y ⫽ ⫺ 共t ⫹ 2兲2

3 54. f 共x兲 ⫽ 3 共x ⫺ 4兲 2 1

55. f 共x兲 ⫽

1 共x2 ⫺ 3x兲2

56. y ⫽

57. g共t兲 ⫽

1 t2 ⫺ 2

58. g共x兲 ⫽

3 ⫺1

3 x3 冪

60. f 共x兲 ⫽ x3共x ⫺ 4兲2

61. y ⫽ x冪2x ⫹ 3

62. y ⫽ t冪t ⫹ 1

63. y ⫽

t 2冪t

64. y ⫽ 冪x 共x ⫺ 2兲

65. y ⫽

冢6x ⫺⫺5x1 冣

2

66. y ⫽

2

冢34x⫺ x冣

3

In Exercises 67–72, find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. Function

Point

36 67. f 共t兲 ⫽ 共3 ⫺ t兲2 68. s共x兲 ⫽

共0, 4兲

1 冪x2 ⫺ 3x ⫹ 4

共3, 12 兲

69. f 共t兲 ⫽ 共t 2 ⫺ 9兲冪t ⫹ 2

共⫺1, ⫺8兲

2x 70. y ⫽ 冪x ⫹ 1

共3, 3兲

x⫹1 71. f 共x兲 ⫽ 冪2x ⫺ 3

共2, 3兲

72. y ⫽

x

共0, 0兲

冪25 ⫹ x2

73. Compound Interest You deposit $1000 in an account with an annual interest rate of r (in decimal form) compounded monthly. At the end of 5 years, the balance is r A ⫽ 1000 1 ⫹ 12





75. Biology The number N of bacteria in a culture after t days is modeled by



N ⫽ 400 1 ⫺

3 . 共t 2 ⫹ 2兲2



Complete the table. What can you conclude? t

0

1

2

3

4

76. Depreciation The value V of a machine t years after it is purchased is inversely proportional to the square root of t ⫹ 1. The initial value of the machine is $10,000. (a) Write V as a function of t.

2

2

173

dN兾dt

冪x ⫹ 2

59. f 共x兲 ⫽ x共3x ⫺ 9兲3 ⫺2

The Chain Rule

60

.

Find the rates of change of A with respect to r when (a) r ⫽ 0.08, (b) r ⫽ 0.10, and (c) r ⫽ 0.12. 74. Environment An environmental study indicates that the average daily level P of a certain pollutant in the air, in parts per million, can be modeled by the equation

(b) Find the rate of depreciation when t ⫽ 1. (c) Find the rate of depreciation when t ⫽ 3. 77. Depreciation Repeat Exercise 76 given that the value of the machine t years after it is purchased is inversely proportional to the cube root of t ⫹ 1. 78. Credit Card Rate The average annual rate r (in percent form) for commercial bank credit cards from 2000 through 2005 can be modeled by r ⫽ 冪⫺1.7409t4 ⫹ 18.070t3 ⫺ 52.68t2 ⫹ 10.9t ⫹ 249 where t represents the year, with t ⫽ 0 corresponding to 2000. (Source: Federal Reserve Bulletin) (a) Find the derivative of this model. Which differentiation rule(s) did you use? (b) Use a graphing utility to graph the derivative on the interval 0 ≤ t ≤ 5. (c) Use the trace feature to find the years during which the finance rate was changing the most. (d) Use the trace feature to find the years during which the finance rate was changing the least. True or False? In Exercises 79 and 80, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. 1 79. If y ⫽ 共1 ⫺ x兲1兾2, then y⬘ ⫽ 2 共1 ⫺ x兲⫺1兾2.

80. If y is a differentiable function of u, u is a differentiable function of v, and v is a differentiable function of x, then dy dy ⫽ dx du

du

dv

⭈ dv ⭈ dx.

81. Given that f 共x) ⫽ h共g共x兲兲, find f⬘共2兲 for each of the following.

P ⫽ 0.25冪0.5n2 ⫹ 5n ⫹ 25

(a) g共2兲 ⫽ ⫺6 and g⬘ 共2兲 ⫽ 5, h共5兲 ⫽ 4 and h⬘ 共⫺6兲 ⫽ 3

where n is the number of residents of the community, in thousands. Find the rate at which the level of pollutant is increasing when the population of the community is 12,000.

(b) g共2兲 ⫽ ⫺1 and g⬘ 共2兲 ⫽ ⫺2, h共2兲 ⫽ 4 and h⬘ 共⫺1兲 ⫽ 5

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174

CHAPTER 2

Differentiation

Section 2.6

Higher-Order Derivatives

■ Find higher-order derivatives. ■ Find and use the position functions to determine the velocity and

acceleration of moving objects.

Second, Third, and Higher-Order Derivatives STUDY TIP In the context of higher-order derivatives, the “standard” derivative f⬘ is often called the first derivative of f.

The derivative of f⬘ is the second derivative of f and is denoted by f ⬙. d 关 f ⬘共x兲兴 ⫽ f ⬙ 共x兲 dx

Second derivative

The derivative of f ⬙ is the third derivative of f and is denoted by f ⬘⬘⬘. d 关 f ⬙共x兲兴 ⫽ f⬘⬘⬘共x兲 dx

Third derivative

By continuing this process, you obtain higher-order derivatives of f. Higherorder derivatives are denoted as follows. D I S C O V E RY

Notation for Higher-Order Derivatives

For each function, find the indicated higher-order derivative.

1. 1st derivative:

y⬘,

f⬘ 共x兲,

a. y ⫽ x2

b. y ⫽ x3

2. 2nd derivative:

y ⬙,

f ⬙ 共x兲,

y⬙

y⬘⬘⬘

3. 3rd derivative:

y⬘⬘⬘,

f ⬘⬘⬘共x兲,

4. 4th derivative:

y 共4兲,

f 共4兲共x兲,

5. nth derivative:

y 共n兲,

f 共n兲共x兲,

c. y ⫽ x 4

d. y ⫽ xn

y 共4兲

y 共n兲

Example 1

dy , dx d 2y , dx 2 d 3y , dx 3 4y d , dx 4 d ny , dx n

d 关 f 共x兲兴, dx d2 关 f 共x兲兴, dx 2 d3 关 f 共x兲兴, dx 3 4 d 关 f 共x兲兴, dx 4 dn 关 f 共x兲兴, dx n

Dx 关 y兴 Dx2 关 y兴 Dx3 关 y兴 Dx4 关 y兴 Dxn 关 y兴

Finding Higher-Order Derivatives

Find the first five derivatives of f 共x兲 ⫽ 2x 4 ⫺ 3x 2. f 共x兲 f⬘ 共x兲 f ⬙ 共x兲 f ⬘⬘⬘共x兲 f 共4兲共x兲 f 共5兲共x兲

⫽ 2x 4 ⫺ 3x 2 ⫽ 8x 3 ⫺ 6x ⫽ 24x 2 ⫺ 6 ⫽ 48x ⫽ 48 ⫽0

Write original function. First derivative Second derivative Third derivative Fourth derivative Fifth derivative

✓CHECKPOINT 1 Find the first four derivatives of f 共x) ⫽ 6x3 ⫺ 2x2 ⫹ 1.



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.6

Example 2

Higher-Order Derivatives

175

Finding Higher-Order Derivatives

Find the value of g⬘⬘⬘共2兲 for the function g共t兲 ⫽ ⫺t 4 ⫹ 2t 3 ⫹ t ⫹ 4. SOLUTION

Original function

Begin by differentiating three times.

g⬘共t兲 ⫽ ⫺4t 3 ⫹ 6t 2 ⫹ 1 g⬙ 共t兲 ⫽ ⫺12t 2 ⫹ 12t g⬘⬘⬘ 共t兲 ⫽ ⫺24t ⫹ 12

First derivative Second derivative Third derivative

Then, evaluate the third derivative of g at t ⫽ 2. g⬘⬘⬘共2兲 ⫽ ⫺24共2兲 ⫹ 12 ⫽ ⫺36

TECHNOLOGY Higher-order derivatives of nonpolynomial functions can be difficult to find by hand. If you have access to a symbolic differentiation utility, try using it to find higher-order derivatives.

Value of third derivative

✓CHECKPOINT 2 Find the value of g⬙⬘共1兲 for g共x兲 ⫽ x 4 ⫺ x3 ⫹ 2x.



Examples 1 and 2 show how to find higher-order derivatives of polynomial functions. Note that with each successive differentiation, the degree of the polynomial drops by one. Eventually, higher-order derivatives of polynomial functions degenerate to a constant function. Specifically, the nth-order derivative of an nth-degree polynomial function f 共x兲 ⫽ an x n ⫹ an⫺1 xn⫺1 ⫹ . . . ⫹ a1x ⫹ a 0 is the constant function f 共n兲共x兲 ⫽ n!an where n! ⫽ 1 ⭈ 2 ⭈ 3 . . . n. Each derivative of order higher than n is the zero function. Polynomial functions are the only functions with this characteristic. For other functions, successive differentiation never produces a constant function.

Example 3

Finding Higher-Order Derivatives

Find the first four derivatives of y ⫽ x⫺1. y ⫽ x ⫺1 ⫽

1 x

y⬘ ⫽ 共⫺1兲x⫺2 ⫽ ⫺

Write original function.

1 x2

y⬙ ⫽ 共⫺1兲共⫺2兲x⫺3 ⫽

✓CHECKPOINT 3 Find the fourth derivative of y⫽

1 . x2

First derivative

2 x3

y⬘⬘⬘ ⫽ 共⫺1兲共⫺2兲共⫺3兲x⫺4 ⫽ ⫺

Second derivative

6 x4

y 共4兲 ⫽ 共⫺1兲共⫺2兲共⫺3兲共⫺4兲x⫺5 ⫽ ■

Third derivative

24 x5

Fourth derivative

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176

CHAPTER 2

Differentiation

Acceleration STUDY TIP Acceleration is measured in units of length per unit of time squared. For instance, if the velocity is measured in feet per second, then the acceleration is measured in “feet per second squared,” or, more formally, in “feet per second per second.”

In Section 2.3, you saw that the velocity of an object moving in a straight path (neglecting air resistance) is given by the derivative of its position function. In other words, the rate of change of the position with respect to time is defined to be the velocity. In a similar way, the rate of change of the velocity with respect to time is defined to be the acceleration of the object. s ⫽ f 共t兲

Position function

ds ⫽ f ⬘共t兲 dt

Velocity function

d 2s ⫽ f ⬙ 共t兲 dt 2

Acceleration function

To find the position, velocity, or acceleration at a particular time t, substitute the given value of t into the appropriate function, as illustrated in Example 4.

Example 4

Finding Acceleration

A ball is thrown upward from the top of a 160-foot cliff, as shown in Figure 2.31. The initial velocity of the ball is 48 feet per second, which implies that the position function is s ⫽ ⫺16t 2 ⫹ 48t ⫹ 160 160 ft

where the time t is measured in seconds. Find the height, the velocity, and the acceleration of the ball when t ⫽ 3. SOLUTION

Not drawn to scale

FIGURE 2.31

Begin by differentiating to find the velocity and acceleration

functions. s ⫽ ⫺16t 2 ⫹ 48t ⫹ 160 ds ⫽ ⫺32t ⫹ 48 dt d 2s ⫽ ⫺32 dt 2

Position function Velocity function Acceleration function

To find the height, velocity, and acceleration when t ⫽ 3, substitute t ⫽ 3 into each of the functions above. Height ⫽ ⫺16共3兲2 ⫹ 48共3兲 ⫹ 160 ⫽ 160 feet Velocity ⫽ ⫺32共3兲 ⫹ 48 ⫽ ⫺48 feet per second Acceleration ⫽ ⫺32 feet per second squared

✓CHECKPOINT 4 A ball is thrown upward from the top of an 80-foot cliff with an initial velocity of 64 feet per second. Give the position function. Then find the velocity and acceleration functions. ■

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SECTION 2.6

Higher-Order Derivatives

177

In Example 4, notice that the acceleration of the ball is ⫺32 feet per second squared at any time t. This constant acceleration is due to the gravitational force of Earth and is called the acceleration due to gravity. Note that the negative value indicates that the ball is being pulled down—toward Earth. Although the acceleration exerted on a falling object is relatively constant near Earth’s surface, it varies greatly throughout our solar system. Large planets exert a much greater gravitational pull than do small planets or moons. The next example describes the motion of a free-falling object on the moon.

Example 5 NASA

The acceleration due to gravity on the surface of the moon is only about one-sixth that exerted by Earth. So, if you were on the moon and threw an object into the air, it would rise to a greater height than it would on Earth’s surface.

An astronaut standing on the surface of the moon throws a rock into the air. The height s (in feet) of the rock is given by s⫽⫺

27 2 t ⫹ 27t ⫹ 6 10

where t is measured in seconds. How does the acceleration due to gravity on the moon compare with that on Earth? SOLUTION

s⫽⫺

✓CHECKPOINT 5 The position function on Earth, where s is measured in meters, t is measured in seconds, v0 is the initial velocity in meters per second, and h0 is the initial height in meters, is s ⫽ ⫺4.9t2 ⫹ v0 t ⫹ h0. If the initial velocity is 2.2 and the initial height is 3.6, what is the acceleration due to gravity on Earth in meters per second per second? ■

Finding Acceleration on the Moon

27 2 t ⫹ 27t ⫹ 6 10

ds 27 ⫽ ⫺ t ⫹ 27 dt 5 2s 27 d ⫽⫺ dt 2 5

Position function Velocity function Acceleration function

So, the acceleration at any time is ⫺

27 ⫽ ⫺5.4 feet per second squared 5

—about one-sixth of the acceleration due to gravity on Earth. The position function described in Example 5 neglects air resistance, which is appropriate because the moon has no atmosphere—and no air resistance. This means that the position function for any free-falling object on the moon is given by s⫽⫺

27 2 t ⫹ v0 t ⫹ h0 10

where s is the height (in feet), t is the time (in seconds), v0 is the initial velocity, and h0 is the initial height. For instance, the rock in Example 5 was thrown upward with an initial velocity of 27 feet per second and had an initial height of 6 feet. This position function is valid for all objects, whether heavy ones such as hammers or light ones such as feathers. In 1971, astronaut David R. Scott demonstrated the lack of atmosphere on the moon by dropping a hammer and a feather from the same height. Both took exactly the same time to fall to the ground. If they were dropped from a height of 6 feet, how long did each take to hit the ground?

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178

CHAPTER 2

Differentiation

Example 6

Finding Velocity and Acceleration

The velocity v (in feet per second) of a certain automobile starting from rest is v⫽

80t t⫹5

Velocity function

where t is the time (in seconds). The positions of the automobile at 10-second intervals are shown in Figure 2.32. Find the velocity and acceleration of the automobile at 10-second intervals from t ⫽ 0 to t ⫽ 60. t=0

t = 10

t = 20

t = 30

t = 40

t = 50

t = 60

FIGURE 2.32

To find the acceleration function, differentiate the velocity function.

SOLUTION

dv 共t ⫹ 5兲共80兲 ⫺ 共80t兲共1兲 ⫽ dt 共t ⫹ 5兲2 ⫽

✓CHECKPOINT 6 Use a graphing utility to graph the velocity function and acceleration function in Example 6 in the same viewing window. Compare the graphs with the table at the right. As the velocity levels off, what does the acceleration approach? ■

400 共t ⫹ 5兲2

Acceleration function

t (seconds)

0

10

20

30

40

50

60

v (ft/sec)

0

53.5

64.0

68.6

71.1

72.7

73.8

dv 共ft兾sec2兲 dt

16

1.78

0.64

0.33

0.20

0.13

0.09

In the table, note that the acceleration approaches zero as the velocity levels off. This observation should agree with your experience—when riding in an accelerating automobile, you do not feel the velocity, but you do feel the acceleration. In other words, you feel changes in velocity.

CONCEPT CHECK 1. Use mathematical notation to write the third derivative of f 冇x冈. 2. Give a verbal description of what is meant by

d 2y . dx 2

3. Complete the following: If f 冇x冈 is an nth-degree polynomial, then f 冇n11冈冇x冈 is equal to ______. 4. If the velocity of an object is constant, what is its acceleration?

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SECTION 2.6

Skills Review 2.6

Higher-Order Derivatives

179

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 1.4 and 2.5.

In Exercises 1–4, solve the equation. 1. ⫺16t 2 ⫹ 24t ⫽ 0

2. ⫺16t2 ⫹ 80t ⫹ 224 ⫽ 0

3. ⫺16t 2 ⫹ 128t ⫹ 320 ⫽ 0

4. ⫺16t 2 ⫹ 9t ⫹ 1440 ⫽ 0

In Exercises 5– 8, find dy/dx. 5. y ⫽ x2共2x ⫹ 7兲

6. y ⫽ 共x 2 ⫹ 3x兲共2x 2 ⫺ 5兲

x2 2x ⫹ 7

7. y ⫽

8. y ⫽

x 2 ⫹ 3x 2x 2 ⫺ 5

In Exercises 9 and 10, find the domain and range of f. 9. f 共x兲 ⫽ x 2 ⫺ 4

10. f 共x兲 ⫽ 冪x ⫺ 7

Exercises 2.6

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–16, find the second derivative of the function.

Function

Value

27. f 共x兲 ⫽

x2

28. g共x兲 ⫽

2x3



3x2

⫹ 3x ⫺ 4兲

f ⬘⬘⬘共⫺2兲

x2

⫺ 5x ⫹ 4兲

g⬘⬘⬘共0兲

1. f 共x兲 ⫽ 9 ⫺ 2x

2. f 共x兲 ⫽ 4x ⫹ 15

3. f 共x兲 ⫽ x 2 ⫹ 7x ⫺ 4

4. f 共x兲 ⫽ 3x 2 ⫹ 4x

1 5. g共t兲 ⫽ 3t 3 ⫺ 4t 2 ⫹ 2t

6. f 共x兲 ⫽ 4共x 2 ⫺ 1兲2

In Exercises 29–34, find the higher-order derivative.

8. g共t兲 ⫽ 32t ⫺2

29. f⬘共x兲 ⫽

3 4t 2

7. f 共t兲 ⫽

9. f 共x兲 ⫽ 3共2 ⫺

x2 3



11. y ⫽ 共x3 ⫺ 2x兲4 13 f 共x兲 ⫽ 15. y ⫽

x⫹1 x⫺1

x2



x2

⫹ 4x ⫹ 8兲

10. f 共x兲 ⫽

3 x x冪

12. y ⫽ 4共x2 ⫹ 5x兲3 14. g共t兲 ⫽ ⫺ 16. h共s兲 ⫽

s3

4 共t ⫹ 2兲2



s2

⫺ 2s ⫹ 1兲



Given

Derivative 2x 2

f ⬙ 共x兲

30. f ⬙ 共x兲 ⫽ 20x 3 ⫺ 36x 2

f ⬘⬘⬘共x兲

31. f⬘⬙ 共x兲 ⫽ 共3x ⫺ 1兲兾x

f 共4兲共x兲

32. f ⬘⬘⬘共x兲 ⫽ 2冪x ⫺ 1

f 共4兲共x兲

33. f 共4兲共x兲 ⫽ 共x2 ⫹ 1兲2

f 共6兲共x兲

34. f⬙ 共x兲 ⫽ 2x2 ⫹ 7x ⫺ 12

f 共5兲共x兲

In Exercises 17–22, find the third derivative of the function.

In Exercises 35–42, find the second derivative and solve the equation f⬙ 冇x冈 ⴝ 0.

17. f 共x兲 ⫽ x 5 ⫺ 3x 4

18. f 共x兲 ⫽ x 4 ⫺ 2x 3

35. f 共x兲 ⫽ x 3 ⫺ 9x 2 ⫹ 27x ⫺ 27

19. f 共x兲 ⫽ 5x共x ⫹ 4兲3

20. f 共x) ⫽ 共x3 ⫺ 6兲4

36. f 共x兲 ⫽ 3x 3 ⫺ 9x ⫹ 1

21. f 共x兲 ⫽

3 16x 2

22. f 共x兲 ⫽

1 x

In Exercises 23–28, find the given value. Function 23. g共t兲 ⫽ 5t 4 ⫹ 10t 2 ⫹ 3 24. f 共x兲 ⫽ 9 ⫺

x2

25. f 共x兲 ⫽ 冪4 ⫺ x 26. f 共t兲 ⫽ 冪2t ⫹ 3

37. f 共x兲 ⫽ 共x ⫹ 3兲共x ⫺ 4兲共x ⫹ 5兲 38. f 共x兲 ⫽ 共x ⫹ 2兲共x ⫺ 2兲共x ⫹ 3兲共x ⫺ 3兲 39. f 共x兲 ⫽ x冪x 2 ⫺ 1

Value

40. f 共x兲 ⫽ x冪4 ⫺ x 2

g⬙ 共2兲

41. f 共x兲 ⫽

x x2 ⫹ 3

f ⬘⬘⬘共⫺5兲

42. f 共x兲 ⫽

x x⫺1

f ⬙ 共⫺冪5 兲 f ⬘⬘⬘ 共12 兲

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180

CHAPTER 2

Differentiation

43. Velocity and Acceleration A ball is propelled straight upward from ground level with an initial velocity of 144 feet per second. (a) Write the position, velocity, and acceleration functions of the ball. (b) When is the ball at its highest point? How high is this point? (c) How fast is the ball traveling when it hits the ground? How is this speed related to the initial velocity? 44. Velocity and Acceleration A brick becomes dislodged from the top of the Empire State Building (at a height of 1250 feet) and falls to the sidewalk below. (a) Write the position, velocity, and acceleration functions of the brick. (b) How long does it take the brick to hit the sidewalk?

51. Modeling Data The table shows the retail values y (in billions of dollars) of motor homes sold in the United States for 2000 to 2005, where t is the year, with t ⫽ 0 corresponding to 2000. (Source: Recreation Vehicle Industry Association) t

0

1

2

3

4

5

y

9.5

8.6

11.0

12.1

14.7

14.4

(a) Use a graphing utility to find a cubic model for the total retail value y共t兲 of the motor homes. (b) Use a graphing utility to graph the model and plot the data in the same viewing window. How well does the model fit the data? (c) Find the first and second derivatives of the function.

(c) How fast is the brick traveling when it hits the sidewalk?

(d) Show that the retail value of motor homes was increasing from 2001 to 2004.

45. Velocity and Acceleration The velocity (in feet per second) of an automobile starting from rest is modeled by

(e) Find the year when the retail value was increasing at the greatest rate by solving y⬙ 共t兲 ⫽ 0.

ds 90t . ⫽ dt t ⫹ 10

(f) Explain the relationship among your answers for parts (c), (d), and (e).

Create a table showing the velocity and acceleration at 10-second intervals during the first minute of travel. What can you conclude?

52. Projectile Motion An object is thrown upward from the top of a 64-foot building with an initial velocity of 48 feet per second.

46. Stopping Distance A car is traveling at a rate of 66 feet per second (45 miles per hour) when the brakes are applied. The position function for the car is given by s ⫽ ⫺8.25t 2 ⫹ 66t, where s is measured in feet and t is measured in seconds. Create a table showing the position, velocity, and acceleration for each given value of t. What can you conclude?

(a) Write the position, velocity, and acceleration functions of the object. (b) When will the object hit the ground? (c) When is the velocity of the object zero? (d) How high does the object go? (e) Use a graphing utility to graph the position, velocity, and acceleration functions in the same viewing window. Write a short paragraph that describes the relationship among these functions.

In Exercises 47 and 48, use a graphing utility to graph f, f⬘, and f⬙ in the same viewing window. What is the relationship among the degree of f and the degrees of its successive derivatives? In general, what is the relationship among the degree of a polynomial function and the degrees of its successive derivatives?

True or False? In Exercises 53–56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

47. f 共x兲 ⫽ x 2 ⫺ 6x ⫹ 6

53. If y ⫽ f 共x兲g共x兲, then y⬘ ⫽ f⬘共x兲g⬘共x兲.

48. f 共x兲 ⫽ 3x 3 ⫺ 9x

In Exercises 49 and 50, the graphs of f, f⬘, and f⬙ are shown on the same set of coordinate axes. Which is which? Explain your reasoning. y

49.

y

50.

−1

x 2

−1

x −1 −2

d 5y ⫽ 0. dx 5

55. If f⬘共c兲 and g⬘共c兲 are zero and h共x兲 ⫽ f 共x兲g共x兲, then h⬘共c兲 ⫽ 0. 56. The second derivative represents the rate of change of the first derivative.

2

−2

54. If y ⫽ 共x ⫹ 1兲共x ⫹ 2兲共x ⫹ 3兲共x ⫹ 4兲, then

3

57. Finding a Pattern Develop a general rule for 关x f 共x兲兴共n兲 where f is a differentiable function of x. 58. Extended Application To work an extended application analyzing the median prices of new privately owned U.S. homes in the South for 1980 through 2005, visit this text’s website at college.hmco.com. (Data Source: U.S. Census Bureau)

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SECTION 2.7

Implicit Differentiation

181

Section 2.7

Implicit Differentiation

■ Find derivatives explicitly. ■ Find derivatives implicitly. ■ Use derivatives to answer questions about real-life situations.

Explicit and Implicit Functions So far in this text, most functions involving two variables have been expressed in the explicit form y ⫽ f 共x兲. That is, one of the two variables has been explicitly given in terms of the other. For example, in the equation y ⫽ 3x ⫺ 5

Explicit form

the variable y is explicitly written as a function of x. Some functions, however, are not given explicitly and are only implied by a given equation, as shown in Example 1.

Example 1

Finding a Derivative Explicitly

Find dy兾dx for the equation xy ⫽ 1. SOLUTION In this equation, y is implicitly defined as a function of x. One way to find dy兾dx is first to solve the equation for y, then differentiate as usual.

xy ⫽ 1

Write original equation.

1 x ⫽ x ⫺1

y⫽

dy ⫽ ⫺x⫺2 dx ⫽⫺

1 x2

Solve for y. Rewrite. Differentiate with respect to x. Simplify.

✓CHECKPOINT 1 Find dy兾dx for the equation x2 y ⫽ 1.



The procedure shown in Example 1 works well whenever you can easily write the given function explicitly. You cannot, however, use this procedure when you are unable to solve for y as a function of x. For instance, how would you find dy兾dx in the equation x 2 ⫺ 2y 3 ⫹ 4y ⫽ 2 where it is very difficult to express y as a function of x explicitly? To do this, you can use a procedure called implicit differentiation.

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182

CHAPTER 2

Differentiation

Implicit Differentiation To understand how to find dy兾dx implicitly, you must realize that the differentiation is taking place with respect to x. This means that when you differentiate terms involving x alone, you can differentiate as usual. But when you differentiate terms involving y, you must apply the Chain Rule because you are assuming that y is defined implicitly as a differentiable function of x. Study the next example carefully. Note in particular how the Chain Rule is used to introduce the dy兾dx factors in Examples 2(b) and 2(d).

Example 2

Applying the Chain Rule

Differentiate each expression with respect to x. a. 3x 2

b. 2y 3

c. x ⫹ 3y

d. xy 2

SOLUTION

a. The only variable in this expression is x. So, to differentiate with respect to x, you can use the Simple Power Rule and the Constant Multiple Rule to obtain d 关3x 2兴 ⫽ 6x. dx b. This case is different. The variable in the expression is y, and yet you are asked to differentiate with respect to x. To do this, assume that y is a differentiable function of x and use the Chain Rule. cu n

c

n

u n⫺1

u⬘

d 关2y3兴 ⫽ dx

2

共3兲

y2

dy dx

⫽ 6y 2

Chain Rule

dy dx

c. This expression involves both x and y. By the Sum Rule and the Constant Multiple Rule, you can write d dy 关x ⫹ 3y兴 ⫽ 1 ⫹ 3 . dx dx d. By the Product Rule and the Chain Rule, you can write d d d 关xy2兴 ⫽ x 关 y 2兴 ⫹ y2 关x兴 dx dx dx

Product Rule

冢 dydx冣 ⫹ y 共1兲

⫽ x 2y ⫽ 2xy

2

Chain Rule

dy ⫹ y 2. dx

✓CHECKPOINT 2 Differentiate each expression with respect to x. a. 4x3

b. 3y2

c. x ⫹ 5y

d. xy3



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SECTION 2.7

Implicit Differentiation

183

Implicit Differentiation

Consider an equation involving x and y in which y is a differentiable function of x. You can use the steps below to find dy兾dx. 1. Differentiate both sides of the equation with respect to x. 2. Write the result so that all terms involving dy兾dx are on the left side of the equation and all other terms are on the right side of the equation. 3. Factor dy兾dx out of the terms on the left side of the equation. 4. Solve for dy兾dx by dividing both sides of the equation by the left-hand factor that does not contain dy兾dx. In Example 3, note that implicit differentiation can produce an expression for dy兾dx that contains both x and y.

Example 3 y=

y

1 2

Find the slope of the tangent line to the ellipse given by x 2 ⫹ 4y 2 ⫽ 4 at the point 共冪2, ⫺1兾冪2 兲, as shown in Figure 2.33.

Ellipse: x 2 + 4y 2 = 4

4 − x2 1

Finding the Slope of a Graph Implicitly

SOLUTION −2

−1

1

(

−1

y = − 12

2, −

1 2

x

x 2 ⫹ 4y 2 ⫽ 4

(

d 2 d 关x ⫹ 4y 2兴 ⫽ 关4兴 dx dx dy 2x ⫹ 8y ⫽0 dx dy 8y ⫽ ⫺2x dx

4 − x2

FIGURE 2.33

冢 冣 冢 冣

Slope of tangent

1 line is 2.

✓CHECKPOINT 3 Find the slope of the tangent line to the circle x2 ⫹ y2 ⫽ 25 at the point 共3, ⫺4兲. y

y=

25 − x2

Circle: x 2 + y 2 = 25

6

2 −4

−2

2

4

6

x

y=−

−6 25 − x2

Implicit differentiation Subtract 2x from each side.

dy ⫺2x ⫽ dx 8y

Divide each side by 8y.

dy x ⫽⫺ dx 4y

Simplify.

To find the slope at the given point, substitute x ⫽ 冪2 and y ⫽ ⫺1兾冪2 into the derivative, as shown below. ⫺

冪2 1 ⫽ 4 共⫺1兾冪2 兲 2

1 y ⫽ ⫺ 冪4 ⫺ x 2 . 2

−2 −4

Differentiate with respect to x.

STUDY TIP To see the benefit of implicit differentiation, try reworking Example 3 using the explicit function

4

−6

Write original equation.

(3, − 4)

The graph of this function is the lower half of the ellipse. ■

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184

CHAPTER 2

Differentiation

Example 4

Using Implicit Differentiation

Find dy兾dx for the equation y 3 ⫹ y 2 ⫺ 5y ⫺ x2 ⫽ ⫺4. SOLUTION

y 3 ⫹ y 2 ⫺ 5y ⫺ x2 ⫽ ⫺4 y 2

(1, 1) (2, 0)

1 −3

−2

−1

1

2

3

x

d d 3 关 y ⫹ y 2 ⫺ 5y ⫺ x2兴 ⫽ 关⫺4兴 dx dx dy dy dy 3y 2 ⫹ 2y ⫺ 5 ⫺ 2x ⫽ 0 dx dx dx dy dy dy 3y 2 ⫹ 2y ⫺ 5 ⫽ 2x dx dx dx

−1

dy 2 共3y ⫹ 2y ⫺ 5兲 ⫽ 2x dx

−2

Write original equation. Differentiate with respect to x. Implicit differentiation Collect dy兾dx terms. Factor.

dy 2x ⫽ dx 3y 2 ⫹ 2y ⫺ 5

(1, − 3) y 3 + y 2 − 5y − x 2 = − 4

The graph of the original equation is shown in Figure 2.34. What are the slopes of the graph at the points 共1, ⫺3兲, 共2, 0兲, and 共1, 1兲?

FIGURE 2.34

✓CHECKPOINT 4 Find dy兾dx for the equation y2 ⫹ x2 ⫺ 2y ⫺ 4x ⫽ 4.

Example 5



Finding the Slope of a Graph Implicitly

Find the slope of the graph of 2x 2 ⫺ y 2 ⫽ 1 at the point 共1, 1兲. SOLUTION

2x 2 − y 2 = 1

Begin by finding dy兾dx implicitly.

2x2 ⫺ y 2 ⫽ 1 dy 4x ⫺ 2y ⫽0 dx dy ⫺2y ⫽ ⫺4x dx

冢 冣 冢 冣

y

dy 2x ⫽ dx y

4 3

−4 − 3 −2

Differentiate with respect to x. Subtract 4x from each side. Divide each side by ⫺2y.

At the point 共1, 1兲, the slope of the graph is

2 1

Write original equation.

(1, 1) 2

3

4

x

2共1兲 ⫽2 1 as shown in Figure 2.35. The graph is called a hyperbola.

−3

✓CHECKPOINT 5

−4

FIGURE 2.35

Hyperbola

Find the slope of the graph of x 2 ⫺ 9y 2 ⫽ 16 at the point 共5, 1兲.



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SECTION 2.7

Implicit Differentiation

185

Application

p

Price (in dollars)

3

Example 6

Demand Function

The demand function for a product is modeled by

(0, 3)

p⫽ 2

0.000001x 3

3 ⫹ 0.01x ⫹ 1

where p is measured in dollars and x is measured in thousands of units, as shown in Figure 2.36. Find the rate of change of the demand x with respect to the price p when x ⫽ 100.

(100, 1)

1

Using a Demand Function

50 100 150 200 250 Demand (in thousands of units)

x

To simplify the differentiation, begin by rewriting the function. Then, differentiate with respect to p. SOLUTION

p⫽

FIGURE 2.36

0.000001x3 ⫹ 0.01x ⫹ 1 ⫽

3 0.000001x 3 ⫹ 0.01x ⫹ 1 3 p

dx dx 3 ⫹ 0.01 ⫽ ⫺ 2 dp dp p dx 3 共0.000003x2 ⫹ 0.01兲 ⫽ ⫺ 2 dp p

0.000003x2

dx 3 ⫽⫺ 2 dp p 共0.000003x2 ⫹ 0.01兲 When x ⫽ 100, the price is p⫽

✓CHECKPOINT 6 The demand function for a product is given by 2 p⫽ . 0.001x2 ⫹ x ⫹ 1 Find dx兾dp implicitly.



3 ⫽ $1. 0.000001共100兲 ⫹ 0.01共100兲 ⫹ 1 3

So, when x ⫽ 100 and p ⫽ 1, the rate of change of the demand with respect to the price is ⫺

3 ⫽ ⫺75. 共1兲2 关0.000003共100兲2 ⫹ 0.01兴

This means that when x ⫽ 100, the demand is dropping at the rate of 75 thousand units for each dollar increase in price.

CONCEPT CHECK 1. Complete the following: The equation x 1 y ⴝ 1 is written in ______ form and the equation y ⴝ 1 ⴚ x is written in ______ form. 2. Complete the following: When you are asked to find dy/dt, you are being asked to find the derivative of ______ with respect to ______. 3. Describe the difference between the explicit form of a function and an implicit equation. Give an example of each. 4. In your own words, state the guidelines for implicit differentiation.

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186

CHAPTER 2

Differentiation

Skills Review 2.7

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Section 0.3.

In Exercises 1– 6, solve the equation for y. 1. x ⫺

y ⫽2 x

2.

4. 12 ⫹ 3y ⫽ 4x2 ⫹ x2y

4 1 ⫽ x⫺3 y

3. xy ⫺ x ⫹ 6y ⫽ 6

5. x2 ⫹ y 2 ⫽ 5

6. x ⫽ ± 冪6 ⫺ y 2

In Exercises 7–10, evaluate the expression at the given point. 7.

3x2 ⫺ 4 , 3y 2

9.

5x , 共⫺1, 2兲 3y 2 ⫺ 12y ⫹ 5

8.

共2, 1兲

10.

x2 ⫺ 2 , 共0, ⫺3兲 1⫺y 1 , 共4, 3兲 y 2 ⫺ 2xy ⫹ x2

Exercises 2.7

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–12, find dy/dx. 1. xy ⫽ 4

2. 3x ⫺ y ⫽ 8x 2

3. y ⫽ 1 ⫺ x , 0 ≤ x ≤ 1

3 4. 4x y ⫺ ⫽ 0 y

5. x 2y 2 ⫺ 2x ⫽ 3

6. xy 2 ⫹ 4xy ⫽ 10

7. 4y 2 ⫺ xy ⫽ 2

8. 2xy 3 ⫺ x 2y ⫽ 2

2

2

2y ⫺ x ⫽5 9. 2 y ⫺3 11.

x⫹y ⫽1 2x ⫺ y

2

12.

13. x2 ⫹ y2 ⫽ 16

共0, 4兲

14. x ⫺ y ⫽ 25

共5, 0兲

15. y ⫹ xy ⫽ 4

共⫺5, ⫺1兲

16. x ⫺ y ⫽ 0

共1, 1兲

17. x3 ⫺ xy ⫹ y2 ⫽ 4

共0, ⫺2兲

18. x y ⫹ y x ⫽ ⫺2

共2, ⫺1兲

19. x3 y 3 ⫺ y ⫽ x

共0, 0兲

20. x3 ⫹ y 3 ⫽ 2xy

共1, 1兲

21. x

共16, 25兲

2

2

2

1兾2

⫹y

1兾2

⫽9

y

y

(1, 4)

x

(−1, −1.5)

2x ⫹ y ⫽1 x ⫺ 5y

Point

3

26. 4x 2 ⫹ 2y ⫺ 1 ⫽ 0

xy ⫺ y ⫽1 10. y⫺x

Equation 2

25. 3x2 ⫺ 2y ⫹ 5 ⫽ 0

2

In Exercises 13–24, find dy/dx by implicit differentiation and evaluate the derivative at the given point.

2

In Exercises 25–30, find the slope of the graph at the given point.

22. 冪xy ⫽ x ⫺ 2y

共4, 1兲

23. x2兾3 ⫹ y 2兾3 ⫽ 5

共8, 1兲

24. 共x ⫹ y兲3 ⫽ x3 ⫹ y 3

共⫺1, 1兲

x

27. x2 ⫹ y 2 ⫽ 4

28. 4x 2 ⫹ y2 ⫽ 4 y

y

(0, 2)

x

x

(0, − 2)

30. x2 ⫺ y 3 ⫽ 0

29. 4x2 ⫹ 9y 2 ⫽ 36 y

y

(

5,

4 3

) x

(−1, 1)

x

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SECTION 2.7 In Exercises 31–34, find dy/dx implicitly and explicitly (the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating dy/dx at the point. 31. x2 ⫹ y 2 ⫽ 25 y=

32. 9x2 ⫹ 16y 2 ⫽ 144

25 − x 2

y=

y

2, 3

3 2

x

y= −

x

33. x ⫺ y 2 ⫺ 1 ⫽ 0 y=

y=

x−1

y

x2 + 7 2

x

y=−

y=−

x2 + 7 2

In Exercises 35– 40, find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window.

36.

x2



y2

37.

y2



5x 3

⫽9

Points

共8, 6兲 and 共⫺6, 8兲

共0, 3兲 and 共2, 冪5 兲

共1, 冪5 兲 and 共1, ⫺ 冪5 兲

38. 4xy ⫹ x2 ⫽ 5

共1, 1兲 and 共5, ⫺1兲

39. x3 ⫹ y 3 ⫽ 8

共0, 2兲 and 共2, 0兲

40. y 2 ⫽

x3 4⫺x

0 < x ≤ 200 0 < x ≤ 500

45. Production Let x represent the units of labor and y the capital invested in a manufacturing process. When 135,540 units are produced, the relationship between labor and capital can be modeled by 100x 0.75y 0.25 ⫽ 135,540.

y2 ⫺ 1141.6 ⫽ 24.9099t3 ⫺ 183.045t2 ⫹ 452.79t where t represents the year, with t ⫽ 1 corresponding to 2001. (Source: U.S. Centers for Disease Control and Prevention)

x−1

Equation

x ≥ 0

47. Health: U.S. HIV/AIDS Epidemic The numbers (in thousands) of cases y of HIV/AIDS reported in the years 2001 through 2005 can be modeled by x

35. x2 ⫹ y 2 ⫽ 100

4 0.000001x2 ⫹ 0.05x ⫹ 1

46. Production Repeat Exercise 45(a) by finding the rate of change of y with respect to x when x ⫽ 3000 and y ⫽ 125.

(3, 2)

(2, − 1)

42. p ⫽

x ≥ 0

(b) The model used in the problem is called the CobbDouglas production function. Graph the model on a graphing utility and describe the relationship between labor and capital.

34. 4y 2 ⫺ x2 ⫽ 7

y

2 0.00001x3 ⫹ 0.1x

(a) Find the rate of change of y with respect to x when x ⫽ 1500 and y ⫽ 1000.

144 − 9x 2 4

y=−

25 − x 2

41. p ⫽

冪2002x⫺ x, 500 ⫺ x , 44. p ⫽ 冪 2x

y

187

Demand In Exercises 41– 44, find the rate of change of x with respect to p.

43. p ⫽

144 − 9x 2 4

(−4, 3)

Implicit Differentiation

(a) Use a graphing utility to graph the model and describe the results. (b) Use the graph to estimate the year during which the number of reported cases was increasing at the greatest rate. (c) Complete the table to estimate the year during which the number of reported cases was increasing at the greatest rate. Compare this estimate with your answer in part (b). t

1

2

3

4

5

y y⬘

共2, 2兲 and 共2, ⫺2兲

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188

CHAPTER 2

Differentiation

Section 2.8

Related Rates

■ Examine related variables. ■ Solve related-rate problems.

Related Variables In this section, you will study problems involving variables that are changing with respect to time. If two or more such variables are related to each other, then their rates of change with respect to time are also related. For instance, suppose that x and y are related by the equation y ⫽ 2x. If both variables are changing with respect to time, then their rates of change will also be related. x and y are related.

The rates of change of x and y are related.

y ⫽ 2x

dy dx ⫽2 dt dt

In this simple example, you can see that because y always has twice the value of x, it follows that the rate of change of y with respect to time is always twice the rate of change of x with respect to time.

Example 1

Examining Two Rates That Are Related

The variables x and y are differentiable functions of t and are related by the equation y ⫽ x 2 ⫹ 3. When x ⫽ 1, dx兾dt ⫽ 2. Find dy兾dt when x ⫽ 1. SOLUTION

respect to t.

Use the Chain Rule to differentiate both sides of the equation with

y ⫽ x2 ⫹ 3 d d 关 y兴 ⫽ 关x 2 ⫹ 3兴 dt dt dy dx ⫽ 2x dt dt

Write original equation. Differentiate with respect to t. Apply Chain Rule.

When x ⫽ 1 and dx兾dt ⫽ 2, you have dy ⫽ 2共1兲共2兲 dt ⫽ 4.

✓CHECKPOINT 1 When x ⫽ 1, dx兾dt ⫽ 3. Find dy兾dt when x ⫽ 1 if y ⫽ x3 ⫹ 2.



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SECTION 2.8

Related Rates

189

Solving Related-Rate Problems In Example 1, you were given the mathematical model. Given equation: y ⫽ x 2 ⫹ 3 dx Given rate: ⫽ 2 when x ⫽ 1 dt dy when x ⫽ 1 dt

Find:

In the next example, you are asked to create a similar mathematical model.

Example 2

Changing Area

A pebble is dropped into a calm pool of water, causing ripples in the form of concentric circles, as shown in the photo. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing? The variables r and A are related by the equation for the area of a circle, A ⫽ ␲ r 2. To solve this problem, use the fact that the rate of change of the radius is given by dr兾dt. SOLUTION

© Randy Faris/Corbis

Total area increases as the outer radius increases.

Equation: A ⫽ ␲ r 2 dr Given rate: ⫽ 1 when r ⫽ 4 dt Find:

dA when r ⫽ 4 dt

Using this model, you can proceed as in Example 1. A ⫽ ␲ r2

Write original equation.

d d 关A兴 ⫽ 关␲ r 2兴 dt dt

✓CHECKPOINT 2 If the radius r of the outer ripple in Example 2 is increasing at a rate of 2 feet per second, at what rate is the total area changing when the radius is 3 feet? ■

Differentiate with respect to t.

dA dr ⫽ 2␲ r dt dt

Apply Chain Rule.

When r ⫽ 4 and dr兾dt ⫽ 1, you have dA ⫽ 2␲ 共4兲共1兲 ⫽ 8␲ dt

Substitute 4 for r and 1 for dr兾dt.

When the radius is 4 feet, the area is changing at a rate of 8␲ square feet per second. STUDY TIP In Example 2, note that the radius changes at a constant rate 共dr兾dt ⫽ 1 for all t兲, but the area changes at a nonconstant rate. When r ⫽ 1 ft

When r ⫽ 2 ft

When r ⫽ 3 ft

When r ⫽ 4 ft

dA ⫽ 2␲ ft 2兾sec dt

dA ⫽ 4␲ ft2兾sec dt

dA ⫽ 6␲ ft2兾sec dt

dA ⫽ 8␲ ft2兾sec dt

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190

CHAPTER 2

Differentiation

The solution shown in Example 2 illustrates the steps for solving a relatedrate problem. Guidelines for Solving a Related-Rate Problem

1. Identify all given quantities and all quantities to be determined. If possible, make a sketch and label the quantities. 2. Write an equation that relates all variables whose rates of change are either given or to be determined. 3. Use the Chain Rule to differentiate both sides of the equation with respect to time. 4. Substitute into the resulting equation all known values of the variables and their rates of change. Then solve for the required rate of change.

STUDY TIP Be sure you notice the order of Steps 3 and 4 in the guidelines. Do not substitute the known values for the variables until after you have differentiated.

In Step 2 of the guidelines, note that you must write an equation that relates the given variables. To help you with this step, reference tables that summarize many common formulas are included in the appendices. For instance, the volume of a sphere of radius r is given by the formula 4 V ⫽ ␲r3 3 as listed in Appendix D. The table below shows the mathematical models for some common rates of change that can be used in the first step of the solution of a related-rate problem. Verbal statement

Mathematical model

The velocity of a car after traveling for 1 hour is 50 miles per hour.

x ⫽ distance traveled dx ⫽ 50 when t ⫽ 1 dt

Water is being pumped into a swimming pool at the rate of 10 cubic feet per minute.

V ⫽ volume of water in pool dV ⫽ 10 ft3兾min dt

A population of bacteria is increasing at the rate of 2000 per hour.

x ⫽ number in population dx ⫽ 2000 bacteria per hour dt

Revenue is increasing at the rate of $4000 per month.

R ⫽ revenue dR ⫽ 4000 dollars per month dt

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SECTION 2.8

Example 3

Related Rates

191

Changing Volume

Air is being pumped into a spherical balloon at the rate of 4.5 cubic inches per minute. See Figure 2.37. Find the rate of change of the radius when the radius is 2 inches. Let V represent the volume of the balloon and let r represent the radius. Because the volume is increasing at the rate of 4.5 cubic inches per minute, you know that dV兾dt ⫽ 4.5. An equation that relates V and r is V ⫽ 43␲ r 3. So, the problem can be represented by the model shown below.

SOLUTION

4 Equation: V ⫽ ␲ r 3 3 dV Given rate: ⫽ 4.5 dt Find:

dr when r ⫽ 2 dt

By differentiating the equation, you obtain 4 V ⫽ ␲ r3 3 d d 4 3 关V兴 ⫽ ␲r dt dt 3 dr dV 4 ⫽ ␲ 共3r 2兲 dt 3 dt 1 dV dr ⫽ . 4␲ r 2 dt dt



FIGURE 2.37

Expanding Balloon

Write original equation.



Differentiate with respect to t. Apply Chain Rule. Solve for dr兾dt.

When r ⫽ 2 and dV兾dt ⫽ 4.5, the rate of change of the radius is dr 1 ⫽ 共4.5兲 dt 4␲ 共22兲 ⬇ 0.09 inch per minute.

✓CHECKPOINT 3 If the radius of a spherical balloon increases at a rate of 1.5 inches per minute, find the rate at which the surface area changes when the radius is 6 inches. 共Formula for surface area of a sphere: S ⫽ 4␲ r 2兲 ■

In Example 3, note that the volume is increasing at a constant rate but the radius is increasing at a variable rate. In this particular example, the radius is increasing more and more slowly as t increases. This is illustrated in the table below. t

1

3

5

7

9

11

V ⫽ 4.5t

4.5

13.5

22.5

31.5

40.5

49.5

1.02

1.48

1.75

1.96

2.13

2.28

0.34

0.16

0.12

0.09

0.08

0.07

冪43V␲

t⫽ dr dt

3

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192

CHAPTER 2

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Example 4

Analyzing a Profit Function

A company’s profit P (in dollars) from selling x units of a product can be modeled by P ⫽ 500x ⫺

冢14冣x . 2

Model for profit

The sales are increasing at a rate of 10 units per day. Find the rate of change in the profit (in dollars per day) when 500 units have been sold. Because you are asked to find the rate of change in dollars per day, you should differentiate the given equation with respect to the time t. SOLUTION

冢14冣x dx 1 dx dP ⫽ 500冢 冣 ⫺ 2冢 冣共x兲冢 冣 dt dt 4 dt P ⫽ 500x ⫺

2

Write model for profit. Differentiate with respect to t.

The sales are increasing at a constant rate of 10 units per day, so dx ⫽ 10. dt When x ⫽ 500 units and dx兾dt ⫽ 10, the rate of change in the profit is STUDY TIP In Example 4, note that one of the keys to successful use of calculus in applied problems is the interpretation of a rate of change as a derivative.

1 dP 共500兲共10兲 ⫽ 500共10兲 ⫺ 2 dt 4 ⫽ 5000 ⫺ 2500 ⫽ $2500 per day.

冢冣

Simplify.

The graph of the profit function (in terms of x) is shown in Figure 2.38. Profit Function

P

Profit (in dollars)

250,000 200,000 150,000 100,000 50,000

500

1000 1500 Units of product sold

2000

x

FIGURE 2.38

✓CHECKPOINT 4 Find the rate of change in profit (in dollars per day) when 50 units have been sold, sales have increased at a rate of 10 units per day, and P ⫽ 200x ⫺ 12 x2. ■

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SECTION 2.8

Related Rates

193

Example 5 MAKE A DECISION

Increasing Production

A company is increasing the production of a product at the rate of 200 units per week. The weekly demand function is modeled by p ⫽ 100 ⫺ 0.001x where p is the price per unit and x is the number of units produced in a week. Find the rate of change of the revenue with respect to time when the weekly production is 2000 units. Will the rate of change of the revenue be greater than $20,000 per week? SOLUTION

Equation: R ⫽ xp ⫽ x共100 ⫺ 0.001x兲 ⫽ 100x ⫺ 0.001x 2 dx Given rate: ⫽ 200 dt Find:

dR when x ⫽ 2000 dt

By differentiating the equation, you obtain R ⫽ 100x ⫺ 0.001x 2 d d 关R兴 ⫽ 关100x ⫺ 0.001x 2兴 dt dt dx dR ⫽ 共100 ⫺ 0.002x兲 . dt dt

✓CHECKPOINT 5

Differentiate with respect to t. Apply Chain Rule.

Using x ⫽ 2000 and dx兾dt ⫽ 200, you have

Find the rate of change of revenue with respect to time for the company in Example 5 if the weekly demand function is p ⫽ 150 ⫺ 0.002x.

Write original equation.



dR ⫽ 关100 ⫺ 0.002共2000兲兴共200兲 dt ⫽ $19,200 per week. No, the rate of change of the revenue will not be greater than $20,000 per week.

CONCEPT CHECK 1. Complete the following. Two variables x and y are changing with respect to ______. If x and y are related to each other, then their rates of change with respect to time are also ______. 2. The volume V of an object is a differentiable function of time t. Describe what dV/dt represents. 3. The area A of an object is a differentiable function of time t. Describe what dA/dt represents. 4. In your own words, state the guidelines for solving related-rate problems.

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194

CHAPTER 2

Differentiation

Skills Review 2.8

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Section 2.7.

In Exercises 1–6, write a formula for the given quantity. 1. Area of a circle

2. Volume of a sphere

3. Surface area of a cube

4. Volume of a cube

5. Volume of a cone

6. Area of a triangle

In Exercises 7–10, find dy/dx by implicit differentiation. 7. x 2 ⫹ y 2 ⫽ 9

8. 3xy ⫺ x 2 ⫽ 6

9. x 2 ⫹ 2y ⫹ xy ⫽ 12

10. x ⫹ xy 2 ⫺ y 2 ⫽ xy

Exercises 2.8

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–4, use the given values to find dy/dt and dx/dt. Equation 1. y ⫽ 冪x

2. y ⫽ 2共x2 ⫺ 3x兲

3. xy ⫽ 4

4. x 2 ⫹ y 2 ⫽ 25

Find

Given dx ⫽3 dt

(a)

dy dt

x ⫽ 4,

(b)

dx dt

x ⫽ 25,

(a)

dy dt

x ⫽ 3,

dx ⫽2 dt

(b)

dx dt

x ⫽ 1,

dy ⫽5 dt

(a)

dy dt

x ⫽ 8,

dx ⫽ 10 dt

(b)

dx dt

x ⫽ 1,

dy ⫽ ⫺6 dt

(a)

dy dt

x ⫽ 3, y ⫽ 4,

dx (b) dt

dy ⫽2 dt

dx ⫽8 dt

dy x ⫽ 4, y ⫽ 3, ⫽ ⫺2 dt

5. Area The radius r of a circle is increasing at a rate of 3 inches per minute. Find the rates of change of the area when (a) r ⫽ 6 inches and (b) r ⫽ 24 inches. 6. Volume The radius r of a sphere is increasing at a rate of 3 inches per minute. Find the rates of change of the volume when (a) r ⫽ 6 inches and (b) r ⫽ 24 inches. 7. Area Let A be the area of a circle of radius r that is changing with respect to time. If dr兾dt is constant, is dA兾dt constant? Explain your reasoning. 8. Volume Let V be the volume of a sphere of radius r that is changing with respect to time. If dr兾dt is constant, is dV兾dt constant? Explain your reasoning.

9. Volume A spherical balloon is inflated with gas at a rate of 10 cubic feet per minute. How fast is the radius of the balloon changing at the instant the radius is (a) 1 foot and (b) 2 feet? 10. Volume The radius r of a right circular cone is increasing at a rate of 2 inches per minute. The height h of the cone is related to the radius by h ⫽ 3r. Find the rates of change of the volume when (a) r ⫽ 6 inches and (b) r ⫽ 24 inches. 11. Cost, Revenue, and Profit A company that manufactures sport supplements calculates that its costs and revenue can be modeled by the equations C ⫽ 125,000 ⫹ 0.75x and

R ⫽ 250x ⫺

1 2 x 10

where x is the number of units of sport supplements produced in 1 week. If production in one particular week is 1000 units and is increasing at a rate of 150 units per week, find: (a) the rate at which the cost is changing. (b) the rate at which the revenue is changing. (c) the rate at which the profit is changing. 12. Cost, Revenue, and Profit A company that manufactures pet toys calculates that its costs and revenue can be modeled by the equations C ⫽ 75,000 ⫹ 1.05x and R ⫽ 500x ⫺

x2 25

where x is the number of toys produced in 1 week. If production in one particular week is 5000 toys and is increasing at a rate of 250 toys per week, find: (a) the rate at which the cost is changing. (b) the rate at which the revenue is changing. (c) the rate at which the profit is changing.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 2.8 13. Volume All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters? 14. Surface Area All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the surface area changing when each edge is (a) 1 centimeter and (b) 10 centimeters? 15. Moving Point A point is moving along the graph of y ⫽ x 2 such that dx兾dt is 2 centimeters per minute. Find dy兾dt for each value of x. (a) x ⫽ ⫺3

(b) x ⫽ 0

(c) x ⫽ 1

(a) x ⫽ ⫺2

(b) x ⫽ 2

(c) x ⫽ 0

(d) x ⫽ 10

17. Moving Ladder A 25-foot ladder is leaning against a house (see figure). The base of the ladder is pulled away from the house at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when the base is (a) 7 feet, (b) 15 feet, and (c) 24 feet from the house? 4 ft / sec

r

12 ft

25 ft

13 ft

ft 2 sec Not drawn to scale

Figure for 17

20. Air Traffic Control An airplane flying at an altitude of 6 miles passes directly over a radar antenna (see figure). When the airplane is 10 miles away 共s ⫽ 10兲, the radar detects that the distance s is changing at a rate of 240 miles per hour. What is the speed of the airplane? y

2nd

x s

6 mi

3rd

Figure for 18

18. Boating A boat is pulled by a winch on a dock, and the winch is 12 feet above the deck of the boat (see figure). The winch pulls the rope at a rate of 4 feet per second. Find the speed of the boat when 13 feet of rope is out. What happens to the speed of the boat as it gets closer and closer to the dock? 19. Air Traffic Control An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other. One airplane is 150 miles from the point and has a speed of 450 miles per hour. The other is 200 miles from the point and has a speed of 600 miles per hour. (a) At what rate is the distance between the planes changing? (b) How much time does the controller have to get one of the airplanes on a different flight path?

x

1st

s

(d) x ⫽ 3

16. Moving Point A point is moving along the graph of y ⫽ 1兾共1 ⫹ x 2兲 such that dx兾dt is 2 centimeters per minute. Find dy兾dt for each value of x.

195

Related Rates

x

90 ft Home

Not drawn to scale

Figure for 20

Figure for 21

21. Baseball A (square) baseball diamond has sides that are 90 feet long (see figure). A player 26 feet from third base is running at a speed of 30 feet per second. At what rate is the player’s distance from home plate changing? 22. Advertising Costs A retail sporting goods store estimates that weekly sales S and weekly advertising costs x are related by the equation S ⫽ 2250 ⫹ 50x ⫹ 0.35x 2. The current weekly advertising costs are $1500, and these costs are increasing at a rate of $125 per week. Find the current rate of change of weekly sales. 23. Environment An accident at an oil drilling platform is causing a circular oil slick. The slick is 0.08 foot thick, and when the radius of the slick is 150 feet, the radius is increasing at the rate of 0.5 foot per minute. At what rate (in cubic feet per minute) is oil flowing from the site of the accident? 24. Profit A company is increasing the production of a product at the rate of 25 units per week. The demand and cost functions for the product are given by p ⫽ 50 ⫺ 0.01x and C ⫽ 4000 ⫹ 40x ⫺ 0.02x 2. Find the rate of change of the profit with respect to time when the weekly sales are x ⫽ 800 units. Use a graphing utility to graph the profit function, and use the zoom and trace features of the graphing utility to verify your result. 25. Sales The profit for a product is increasing at a rate of $5600 per week. The demand and cost functions for the product are given by p ⫽ 6000 ⫺ 25x and C ⫽ 2400x ⫹ 5200. Find the rate of change of sales with respect to time when the weekly sales are x ⫽ 44 units. 26. Cost The annual cost (in millions of dollars) for a government agency to seize p% of an illegal drug is given by C⫽

528p , 100 ⫺ p

0 ≤ p < 100.

The agency’s goal is to increase p by 5% per year. Find the rates of change of the cost when (a) p ⫽ 30% and (b) p ⫽ 60%. Use a graphing utility to graph C. What happens to the graph of C as p approaches 100?

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196

CHAPTER 2

Differentiation

Algebra Review Simplifying Algebraic Expressions To be successful in using derivatives, you must be good at simplifying algebraic expressions. Here are some helpful simplification techniques.

TECHNOLOGY Symbolic algebra systems can simplify algebraic expressions. If you have access to such a system, try using it to simplify the expressions in this Algebra Review.

1. Combine like terms. This may involve expanding an expression by multiplying factors. 2. Divide out like factors in the numerator and denominator of an expression. 3. Factor an expression. 4. Rationalize a denominator. 5. Add, subtract, multiply, or divide fractions.

Example 1 a.

Simplifying a Fractional Expression

共x ⫹ ⌬x兲2 ⫺ x 2 x 2 ⫹ 2x共⌬x兲 ⫹ 共⌬x兲2 ⫺ x2 ⫽ ⌬x ⌬x ⫽

2x共⌬x兲 ⫹ 共⌬x兲2 ⌬x

Combine like terms.



⌬x共2x ⫹ ⌬x兲 ⌬x

Factor.

⫽ 2x ⫹ ⌬x, b.

Expand expression.

⌬x ⫽ 0

Divide out like factors.

共x 2 ⫺ 1兲共⫺2 ⫺ 2x兲 ⫺ 共3 ⫺ 2x ⫺ x 2兲共2兲 共x 2 ⫺ 1兲2

c. 2



共⫺2x 2 ⫺ 2x 3 ⫹ 2 ⫹ 2x兲 ⫺ 共6 ⫺ 4x ⫺ 2x 2兲 共x 2 ⫺ 1兲2

Expand expression.



⫺2x 2 ⫺ 2x 3 ⫹ 2 ⫹ 2x ⫺ 6 ⫹ 4x ⫹ 2x 2 共x 2 ⫺ 1兲2

Remove parentheses.



⫺2x 3 ⫹ 6x ⫺ 4 共x 2 ⫺ 1兲2

Combine like terms.

冢2x3x⫹ 1冣冤 3x共2兲 ⫺共3x共2x兲 ⫹ 1兲共3兲冥 2

⫽2

冢2x3x⫹ 1冣冤 6x ⫺共3x共6x兲 ⫹ 3兲冥 2

Multiply factors.



2共2x ⫹ 1兲共6x ⫺ 6x ⫺ 3兲 共3x兲3

Multiply fractions and remove parentheses.



2共2x ⫹ 1兲共⫺3兲 3共9兲x 3

Combine like terms and factor.



⫺2共2x ⫹ 1兲 9x 3

Divide out like factors.

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Algebra Review

Example 2

197

Simplifying an Expression with Powers or Radicals

a. 共2x ⫹ 1兲 2共6x ⫹ 1兲 ⫹ 共3x 2 ⫹ x兲共2兲共2x ⫹ 1兲共2兲 ⫽ 共2x ⫹ 1兲关共2x ⫹ 1兲共6x ⫹ 1兲 ⫹ 共3x 2 ⫹ x兲共2兲共2兲兴

Factor.

⫽ 共2x ⫹ 1兲关

⫹ 8x ⫹ 1 ⫹ 共

Multiply factors.

⫽ 共2x ⫹ 1兲共

⫹ 8x ⫹ 1 ⫹

12x 2 12x 2

12x 2

12x 2

⫹ 4x兲兴

⫹ 4x兲

⫽ 共2x ⫹ 1兲共24x2 ⫹ 12x ⫹ 1兲

Remove parentheses. Combine like terms.

b. 共⫺1兲共6x 2 ⫺ 4x兲⫺2共12x ⫺ 4兲

c. 共x兲



共⫺1兲共12x ⫺ 4兲 共6x 2 ⫺ 4x兲2

Rewrite as a fraction.



共⫺1兲共4兲共3x ⫺ 1兲 共6x 2 ⫺ 4x兲2

Factor.



⫺4共3x ⫺ 1兲 共6x 2 ⫺ 4x兲2

Multiply factors.

冢12冣共2x ⫹ 3兲

⫺1兾2

⫽ 共2x ⫹ 3兲⫺1兾2

d.

⫹ 共2x ⫹ 3兲1兾2共1兲

冢12冣关x ⫹ 共2x ⫹ 3兲共2兲兴

Factor.



x ⫹ 4x ⫹ 6 共2x ⫹ 3兲1兾2共2兲

Rewrite as a fraction.



5x ⫹ 6 2共2x ⫹ 3兲1兾2

Combine like terms.

x 2共12 兲共2x兲共x 2 ⫹ 1兲⫺1兾2 ⫺ 共x 2 ⫹ 1兲1兾2共2x兲 x4 ⫽

共x 3兲共x 2 ⫹ 1兲⫺1兾2 ⫺ 共x 2 ⫹ 1兲1兾2共2x兲 x4

Multiply factors.



共x 2 ⫹ 1兲⫺1兾2共x兲关x 2 ⫺ 共x 2 ⫹ 1兲共2兲兴 x4

Factor.



x关x 2 ⫺ 共2x 2 ⫹ 2兲兴 共x 2 ⫹ 1兲1兾2x 4

Write with positive exponents.



x 2 ⫺ 2x 2 ⫺ 2 共x 2 ⫹ 1兲1兾2x 3

Divide out like factors and remove parentheses.



⫺x 2 ⫺ 2 共x 2 ⫹ 1兲1兾2x 3

Combine like terms.

All but one of the expressions in this Algebra Review are derivatives. Can you see what the original function is? Explain your reasoning.

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198

CHAPTER 2

Differentiation

Chapter Summary and Study Strategies After studying this chapter, you should have acquired the following skills. The exercise numbers are keyed to the Review Exercises that begin on page 200. Answers to odd-numbered Review Exercises are given in the back of the text.*

Section 2.1

Review Exercises 1–4



Approximate the slope of the tangent line to a graph at a point.



Interpret the slope of a graph in a real-life setting.

5–8



Use the limit definition to find the derivative of a function and the slope of a graph at a point.

9–16

f⬘共x兲 ⫽ lim

⌬x→0

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x



Use the derivative to find the slope of a graph at a point.

17–24



Use the graph of a function to recognize points at which the function is not differentiable.

25–28

Section 2.2 ■

Use the Constant Multiple Rule for differentiation.

29, 30

d 关c f 共x兲兴 ⫽ c f⬘共x兲 dx ■

Use the Sum and Difference Rules for differentiation.

31–38

d 关 f 共x兲 ± g共x兲兴 ⫽ f⬘共x兲 ± g⬘共x兲 dx

Section 2.3 ■

Find the average rate of change of a function over an interval and the instantaneous rate of change at a point. Average rate of change ⫽

39, 40

f 共b兲 ⫺ f 共a兲 b⫺a

Instantaneous rate of change ⫽ lim

⌬x→0

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x



Find the average and instantaneous rates of change of a quantity in a real-life problem.

41–44



Find the velocity of an object that is moving in a straight line.

45, 46



Create mathematical models for the revenue, cost, and profit for a product.

47, 48

P ⫽ R ⫺ C, R ⫽ xp ■

Find the marginal revenue, marginal cost, and marginal profit for a product.

49–58

* Use a wide range of valuable study aids to help you master the material in this chapter. The Student Solutions Guide includes step-by-step solutions to all odd-numbered exercises to help you review and prepare. The student website at college.hmco.com/info/larsonapplied offers algebra help and a Graphing Technology Guide. The Graphing Technology Guide contains step-by-step commands and instructions for a wide variety of graphing calculators, including the most recent models.

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Chapter Summary and Study Strategies

Section 2.4 ■

199

Review Exercises

Use the Product Rule for differentiation.

59–62

d 关 f 共x兲g共x兲兴 ⫽ f 共x兲g⬘共x兲 ⫹ g共x兲 f⬘共x兲 dx ■

Use the Quotient Rule for differentiation.

63, 64

g共x兲 f⬘ 共x兲 ⫺ f 共x兲g⬘ 共x兲 d f 共x兲 ⫽ dx g共x兲 关g共x兲兴 2

冤 冥

Section 2.5 ■

Use the General Power Rule for differentiation.

65–68

d n 关u 兴 ⫽ nu n⫺1u⬘ dx ■

Use differentiation rules efficiently to find the derivative of any algebraic function, then simplify the result.

69–78



Use derivatives to answer questions about real-life situations. (Sections 2.1–2.5)

79, 80

Section 2.6 ■

Find higher-order derivatives.

81–88



Find and use the position function to determine the velocity and acceleration of a moving object.

89, 90

Section 2.7 ■

91–98

Find derivatives implicitly.

Section 2.8 ■

99, 100

Solve related-rate problems.

Study Strategies ■

Simplify Your Derivatives Often our students ask if they have to simplify their derivatives. Our answer is “Yes, if you expect to use them.” In the next chapter, you will see that almost all applications of derivatives require that the derivatives be written in simplified form. It is not difficult to see the advantage of a derivative in simplified form. Consider, for instance, the derivative of f 共x兲 ⫽ x兾冪x 2 ⫹ 1. The “raw form” produced by the Quotient and Chain Rules

共x 2 ⫹ 1兲1兾2共1兲 ⫺ 共x兲共2 兲共x 2 ⫹ 1兲⫺1兾2共2x兲 共冪x2 ⫹ 1 兲2 is obviously much more difficult to use than the simplified form 1 . f⬘共x兲 ⫽ 2 共x ⫹ 1兲3兾2 1

f⬘共x兲 ⫽



List Units of Measure in Applied Problems When using derivatives in real-life applications, be sure to list the units of measure for each variable. For instance, if R is measured in dollars and t is measured in years, then the derivative dR兾dt is measured in dollars per year.

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200

CHAPTER 2

Differentiation

Review Exercises

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 4, approximate the slope of the tangent line to the graph at 冇x, y冈. 2.

(x, y) (x, y)

Number of subscribers (in millions)

1.

Cellular Phone Subscribers S 240 200 160 120 80 40 6

7

8

9

10 11 12 13 14 15

t

Year (6 ↔ 1996)

3.

4.

(x, y)

Figure for 6

(x, y)

7. Medicine The graph shows the estimated number of milligrams of a pain medication M in the bloodstream t hours after a 1000-milligram dose of the drug has been given. Estimate the slopes of the graph at t ⫽ 0, 4, and 6. Pain Medication in Bloodstream

5. Sales The graph approximates the annual sales S (in millions of dollars per year) of Home Depot for the years 1999 through 2005, where t is the year, with t ⫽ 9 corresponding to 1999. Estimate the slopes of the graph when t ⫽ 10, t ⫽ 13, and t ⫽ 15. Interpret each slope in the context of the problem. (Source: The Home Depot, Inc.)

Pain medication (in milligrams)

M

1000 800 600 400 200 1

2

4

5

6

7

t

Hours

Home Depot Sales S

8. White-Water Rafting Two white-water rafters leave a campsite simultaneously and start downstream on a 9-mile trip. Their distances from the campsite are given by s ⫽ f 共t兲 and s ⫽ g共t兲, where s is measured in miles and t is measured in hours.

85,000 80,000 75,000 70,000 65,000 60,000 55,000 50,000 45,000 40,000 35,000

White-Water Rafting s 9 10 11 12 13 14 15

t

Year (9 ↔ 1999)

6. Consumer Trends The graph approximates the number of subscribers S (in millions per year) of cellular telephones for the years 1996 through 2005, where t is the year, with t ⫽ 6 corresponding to 1996. Estimate the slopes of the graph when t ⫽ 7, t ⫽ 11, and t ⫽ 15. Interpret each slope in the context of the problem. (Source: Cellular Telecommunications & Internet Association)

Distance (in miles)

Annual sales (in millions of dollars)

3

12 10 8 6 4 2

s = f(t)

s = g(t)

t1 t2 t3

t

Time (in hours)

(a) Which rafter is traveling at a greater rate at t 1? (b) What can you conclude about their rates at t 2? (c) What can you conclude about their rates at t 3? (d) Which rafter finishes the trip first? Explain your reasoning.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Review Exercises In Exercises 9–16, use the limit definition to find the derivative of the function. Then use the limit definition to find the slope of the tangent line to the graph of f at the given point. 9. f 共x兲 ⫽ ⫺3x ⫺ 5; 共⫺2, 1兲 10. f 共x兲 ⫽ 7x ⫹ 3; 共⫺1, 4兲 11. f 共x兲 ⫽

x2

⫺ 4x; 共1, ⫺3兲

12. f 共x兲 ⫽

x2

⫹ 10; 共2, 14兲

31. f 共x兲 ⫽ x 2 ⫹ 3, 共1, 4兲 32. f 共x兲 ⫽ 2x 2 ⫺ 3x ⫹ 1, 共2, 3兲 33. y ⫽ 11x 4 ⫺ 5x 2 ⫹ 1, 共⫺1, 7兲 34. y ⫽ x 3 ⫺ 5 ⫹

3 , x3 1

共⫺1, ⫺9兲

13. f 共x兲 ⫽ 冪x ⫹ 9; 共⫺5, 2兲

14. f 共x兲 ⫽ 冪x ⫺ 1; 共10, 3兲

35. f 共x兲 ⫽ 冪x ⫺

1 ; 共6, 1兲 15. f 共x兲 ⫽ x⫺5

1 ; 共⫺3, 1兲 16. f 共x兲 ⫽ x⫹4

36. f 共x兲 ⫽ 2x⫺3 ⫹ 4 ⫺ 冪x,

In Exercises 17–24, find the slope of the graph of f at the given point. 17. f (x兲 ⫽ 5 ⫺ 3x; 共1, ⫺2兲

18. f 共x兲 ⫽ 1 ⫺ 4x; 共2, ⫺7兲

1 19. f 共x兲 ⫽ ⫺ 2 x 2 ⫹ 2x; 共2, 2兲

20. f 共x兲 ⫽ 4 ⫺ x 2; 共⫺1, 3兲

21. f 共x兲 ⫽ 冪x ⫹ 2; 共9, 5兲

22. f 共x兲 ⫽ 2冪x ⫹ 1; 共4, 5兲

5 23. f 共x兲 ⫽ ; 共1, 5兲 x

24. f 共x兲 ⫽

2 ⫺ 1; x

冢12, 3冣

x⫹1 x⫺1

ⱍⱍ

26. y ⫽ ⫺ x ⫹ 3

y

y

4

4

2

3

2

4

6

x

−2

27. y ⫽

1

2

3

x

x ≤ 0 x > 0

y

y 4 2

1

2

3

x

−4

−2

2

x

−2

−2

In Exercises 29–38, find the equation of the tangent line at the given point. Then use a graphing utility to graph the function and the equation of the tangent line in the same viewing window. 2 29. g共t兲 ⫽ 2, 3t 30. h共x兲 ⫽

2 , 共3x兲 2

冢 冣 2 1, 3

冢2, 181 冣

共⫺4, ⫺4兲

In Exercises 39 and 40, find the average rate of change of the function over the indicated interval. Then compare the average rate of change with the instantaneous rates of change at the endpoints of the interval. 40. f 共x兲 ⫽ x 3 ⫹ x; 关⫺2, 2兴 41. Sales The annual sales S (in millions of dollars per year) of Home Depot for the years 1999 through 2005 can be modeled by

(a) Find the average rate of change for the interval from 1999 through 2005.

S⫽

1 −3 −2

38. f 共x兲 ⫽ ⫺x 2 ⫺ 4x ⫺ 4,

42. Consumer Trends The numbers of subscribers S (in millions per year) of cellular telephones for the years 1996 through 2005 can be modeled by

3 2

x ⫹3 , 共1, 4兲 x

(c) Interpret the results of parts (a) and (b) in the context of the problem.

28. y ⫽ 共x ⫹ 1兲 2兾3

4

共1, 5兲

2

(b) Find the instantaneous rates of change of the model for 1999 and 2005.

−2

3

共1, 0兲

where t is the time in years, with t ⫽ 9 corresponding to 1999. A graph of this model appears in Exercise 5. (Source: The Home Depot, Inc.)

−3 −2 −1

⫺ 2, 冦⫺x x ⫹ 2,

,

S ⫽ 123.833t3 ⫺ 4319.55t2 ⫹ 56,278.0t ⫺ 208,517

1

−4

37. f 共x兲 ⫽

冪x

39. f 共x兲 ⫽ x 2 ⫹ 3x ⫺ 4; 关0, 1兴

In Exercises 25–28, determine the x-value at which the function is not differentiable. 25. y ⫽

201

⫺33.2166 ⫹ 11.6732t 1 ⫺ 0.0207t

where t is the time in years, with t ⫽ 6 corresponding to 1996. A graph of this model appears in Exercise 6. (Source: Cellular Telecommunications & Internet Association) (a) Find the average rate of change for the interval from 2000 through 2005. (b) Find the instantaneous rates of change of the model for 2000 and 2005. (c) Interpret the results of parts (a) and (b) in the context of the problem.

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202

CHAPTER 2

Differentiation

43. Retail Price The average retail price P (in dollars) of a half-gallon of prepackaged ice cream from 1992 through 2006 can be modeled by the equation P ⫽ ⫺0.00149t3 ⫹ 0.0340t2 ⫺ 0.086t ⫹ 2.53 where t is the year, with t ⫽ 2 corresponding to 1992. (Source: U.S. Bureau of Labor Statistics) (a) Find the rate of change of the price with respect to the year. (b) At what rate was the price of a half gallon of prepackaged ice cream changing in 1997? in 2003? in 2005?

47. Cost, Revenue, and Profit The fixed cost of operating a small flower shop is $2500 per month. The average cost of a floral arrangement is $15 and the average price is $27.50. Write the monthly revenue, cost, and profit functions for the floral shop in terms of x, the number of arrangements sold. 48. Profit The weekly demand and cost functions for a product are given by p ⫽ 1.89 ⫺ 0.0083x and

C ⫽ 21 ⫹ 0.65x.

Write the profit function for this product.

(c) Use a graphing utility to graph the function for 2 ≤ t ≤ 16. During which years was the price increasing? decreasing?

Marginal Cost cost function.

(d) For what years do the slopes of the tangent lines appear to be positive? negative?

49. C ⫽ 2500 ⫹ 320x

50. C ⫽ 225x ⫹ 4500

51. C ⫽ 370 ⫹ 2.55冪x

52. C ⫽ 475 ⫹ 5.25x 2兾3

(e) Compare your answers for parts (c) and (d). 44. Recycling The amount T of recycled paper products in millions of tons from 1997 through 2005 can be modeled by the equation T ⫽ 冪1.3150t3 ⫺ 42.747t2 ⫹ 522.28t ⫺ 885.2 where t is the year, with t ⫽ 7 corresponding to 1997. (Source: Franklin Associates, Ltd.) (a) Use a graphing utility to graph the equation. Be sure to choose an appropriate window. (b) Determine dT兾dt. Evaluate dT兾dt for 1997, 2002, and 2005. (c) Is dT兾dt positive for t ≥ 7? Does this agree with the graph of the function? What does this tell you about this situation? Explain your reasoning. 45. Velocity A rock is dropped from a tower on the Brooklyn Bridge, 276 feet above the East River. Let t represent the time in seconds. (a) Write a model for the position function (assume that air resistance is negligible). (b) Find the average velocity during the first 2 seconds. (c) Find the instantaneous velocities when t ⫽ 2 and t ⫽ 3.

In Exercises 49–52, find the marginal

Marginal Revenue In Exercises 53–56, find the marginal revenue function. 1 53. R ⫽ 200x ⫺ x 2 5 55. R ⫽

35x 冪x ⫺ 2

, x ≥ 6

In Exercises 59–78, find the derivative of the function. Simplify your result. State which differentiation rule(s) you used to find the derivative. 59. f 共x兲 ⫽ x 3共5 ⫺ 3x 2兲

60. y ⫽ 共3x 2 ⫹ 7兲共x 2 ⫺ 2x兲

61. y ⫽ 共4x ⫺ 3兲共x 3 ⫺ 2x 2兲

62. s ⫽ 4 ⫺

63. f 共x兲 ⫽

6x ⫺ 5 x2 ⫹ 1

Time, t Velocity

0

2

65. f 共x兲 ⫽ 共5x 2 ⫹ 2兲 3

6

8



1 2 共t ⫺ 3t兲 t2

x ⫹x⫺1 x2 ⫺ 1 2

3 2 x ⫺1 66. f 共x兲 ⫽ 冪

2 冪x ⫹ 1

69. g共x兲 ⫽ x冪x 2 ⫹ 1 70. g共t兲 ⫽

4



64. f 共x兲 ⫽

68. g共x兲 ⫽ 冪x 6 ⫺ 12x 3 ⫹ 9

where t is the time (in seconds). Complete the table, showing the velocity of the bicyclist at two-second intervals.



1 58. P ⫽ ⫺ 15 x 3 ⫹ 4000x 2 ⫺ 120x ⫺ 144,000

(e) When it hits the water, what is the rock’s speed?

0 ≤ t ≤ 8

10 冪x

57. P ⫽ ⫺0.0002x 3 ⫹ 6x 2 ⫺ x ⫺ 2000

67. h共x兲 ⫽

s ⫽ 2t 3兾2,



56. R ⫽ x 5 ⫹

Marginal Profit In Exercises 57 and 58, find the marginal profit function.

(d) How long will it take for the rock to hit the water? 46. Velocity The straight-line distance s (in feet) traveled by an accelerating bicyclist can be modeled by

3 54. R ⫽ 150x ⫺ x2 4

t 共1 ⫺ t兲3

71. f 共x兲 ⫽ x共1 ⫺ 4x 2兲2



72. f 共x兲 ⫽ x 2 ⫹

1 x



5

73. h共x兲 ⫽ 关x 2共2x ⫹ 3兲兴 3

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Review Exercises

90. Velocity and Acceleration The position function of a particle is given by

74. f 共x兲 ⫽ 关共x ⫺ 2兲共x ⫹ 4兲兴 2 75. f 共x兲 ⫽

x2

76. f 共s兲 ⫽

s3

77. h共t兲 ⫽

冪3t ⫹ 1 共1 ⫺ 3t兲2

78. g共x兲 ⫽

共3x ⫹ 1兲2 共x 2 ⫹ 1兲2

共x ⫺ 1兲

5

共s 2 ⫺ 1兲5兾2

s⫽

In Exercises 91–94, use implicit differentiation to find dy/dx.

1300 T⫽ 2 t ⫹ 2t ⫹ 25

91. x 2 ⫹ 3xy ⫹ y 3 ⫽ 10 92. x 2 ⫹ 9xy ⫹ y 2 ⫽ 0 93. y 2 ⫺ x 2 ⫹ 8x ⫺ 9y ⫺ 1 ⫽ 0

where t is the time (in hours). (a) Find the rates of change of T when t ⫽ 1, t ⫽ 3, t ⫽ 5, and t ⫽ 10. (b) Graph the model on a graphing utility and describe the rate at which the temperature is changing. 80. Forestry According to the Doyle Log Rule, the volume V (in board-feet) of a log of length L (feet) and diameter D (inches) at the small end is

冢D ⫺4 4冣 L. 2

Find the rates at which the volume is changing with respect to D for a 12-foot-long log whose smallest diameter is (a) 8 inches, (b) 16 inches, (c) 24 inches, and (d) 36 inches. In Exercises 81– 88, find the higher-order derivative.

94. y 2 ⫹ x 2 ⫺ 6y ⫺ 2x ⫺ 5 ⫽ 0 In Exercises 95–98, use implicit differentiation to find an equation of the tangent line at the given point. Equation

5x 4



6x2

Point

95. y 2 ⫽ x ⫺ y

共2, 1兲

96.

3 x 2冪

共8, 4兲

97.

y2

⫺ 2x ⫽ xy

98.

y3

2x2 y



⫹ 3冪y ⫽ 10 ⫹

共1, 2兲

3xy 2

⫽ ⫺1

ft 3

10 min

⫹ 2x, find f⬘⬘⬘共x兲.

4 ft

6 83. Given f⬘⬘⬘共x兲 ⫽ ⫺ 4, find f 共5兲共x兲. x 84. Given f 共x兲 ⫽ 冪x, find f 共4兲共x兲. 85. Given f⬘共x兲 ⫽ 7x

, find f ⬙ 共x兲.

20 ft 9 ft 40 ft

5兾2

3 86. Given f 共x兲 ⫽ x2 ⫹ , find f ⬙ 共x兲. x 3 x, 87. Given f ⬙ 共x兲 ⫽ 6冪 find f⬘⬘⬘共x兲.

88. Given f⬘⬘⬘共x兲 ⫽ 20x 4 ⫺

2 , find f 共5兲共x兲. x3

89. Athletics A person dives from a 30-foot platform with an initial velocity of 5 feet per second (upward). (a) Find the position function of the diver. (b) How long will it take for the diver to hit the water? (c) What is the diver’s velocity at impact? (d) What is the diver’s acceleration at impact?

共0, ⫺1兲

99. Water Level A swimming pool is 40 feet long, 20 feet wide, 4 feet deep at the shallow end, and 9 feet deep at the deep end (see figure). Water is being pumped into the pool at the rate of 10 cubic feet per minute. How fast is the water level rising when there is 4 feet of water in the deep end?

81. Given f 共x兲 ⫽ 3x 2 ⫹ 7x ⫹ 1, find f ⬙ 共x兲. 82. Given f⬘共x兲 ⫽

1 t 2 ⫹ 2t ⫹ 1

where s is the height (in feet) and t is the time (in seconds). Find the velocity and acceleration functions.

79. Physical Science The temperature T (in degrees Fahrenheit) of food placed in a freezer can be modeled by

V⫽

203

100. Profit The demand and cost functions for a product can be modeled by p ⫽ 211 ⫺ 0.002x and C ⫽ 30x ⫹ 1,500,000 where x is the number of units produced. (a) Write the profit function for this product. (b) Find the marginal profit when 80,000 units are produced. (c) Graph the profit function on a graphing utility and use the graph to determine the price you would charge for the product. Explain your reasoning.

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204

CHAPTER 2

Differentiation

Chapter Test

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Take this test as you would take a test in class. When you are done, check your work against the answers given in the back of the book. In Exercises 1 and 2, use the limit definition to find the derivative of the function. Then find the slope of the tangent line to the graph of f at the given point. 1. f 共x兲 ⫽ x2 ⫹ 1; 共2, 5兲

2. f 共x兲 ⫽ 冪x ⫺ 2; 共4, 0兲

In Exercises 3 –11, find the derivative of the function. Simplify your result. 3. f 共t兲 ⫽ t3 ⫹ 2t

4. f 共x兲 ⫽ 4x2 ⫺ 8x ⫹ 1

5. f 共x兲 ⫽ x3兾2

6. f 共x兲 ⫽ 共x ⫹ 3兲共x ⫺ 3兲

7. f 共x兲 ⫽

8. f 共x兲 ⫽ 冪x 共5 ⫹ x兲

9. f 共x兲 ⫽ 共3x2 ⫹ 4兲2

⫺3x⫺3

10. f 共x兲 ⫽ 冪1 ⫺ 2x

11. f 共x兲 ⫽

共5x ⫺ 1兲3 x

1 at the point 共1, 0兲. x Then use a graphing utility to graph the function and the tangent line in the same viewing window.

12. Find an equation of the tangent line to the graph of f 共x兲 ⫽ x ⫺

13. The annual sales S (in millions of dollars per year) of Bausch & Lomb for the years 1999 through 2005 can be modeled by S ⫽ ⫺2.9667t 3 ⫹ 135.008t 2 ⫺ 1824.42t ⫹ 9426.3, 9 ≤ t ≤ 15 where t represents the year, with t ⫽ 9 corresponding to 1999. Lomb, Inc.)

(Source: Bausch &

(a) Find the average rate of change for the interval from 2001 through 2005. (b) Find the instantaneous rates of change of the model for 2001 and 2005. (c) Interpret the results of parts (a) and (b) in the context of the problem. 14. The monthly demand and cost functions for a product are given by p ⫽ 1700 ⫺ 0.016x

and

C ⫽ 715,000 ⫹ 240x.

Write the profit function for this product. In Exercises 15–17, find the third derivative of the function. Simplify your result. 15. f 共x兲 ⫽ 2x2 ⫹ 3x ⫹ 1

16. f 共x兲 ⫽ 冪3 ⫺ x

17. f 共x兲 ⫽

2x ⫹ 1 2x ⫺ 1

In Exercises 18–20, use implicit differentiation to find dy/dx. 18. x ⫹ xy ⫽ 6

19. y2 ⫹ 2x ⫺ 2y ⫹ 1 ⫽ 0

20. x2 ⫺ 2y2 ⫽ 4

21. The radius r of a right circular cylinder is increasing at a rate of 0.25 centimeter per minute. The height h of the cylinder is related to the radius by h ⫽ 20r. Find the rate of change of the volume when (a) r ⫽ 0.5 centimeter and (b) r ⫽ 1 centimeter.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

3

Still Images/Getty Images

Applications of the Derivative

Designers use the derivative to find the dimensions of a container that will minimize cost. (See Section 3.4, Exercise 28.)

3.1 3.2

Applications Derivatives have many real-life applications. The applications listed below represent a sample of the applications in this chapter. ■ ■ ■ ■ ■

Profit Analysis, Exercise 43, page 214 Phishing, Exercise 75, page 234 Average Cost, Exercises 61 and 62, page 265 Make a Decision: Social Security, Exercise 55, page 274 Economics: Gross Domestic Product, Exercise 41, page 282

3.3

3.4 3.5

3.6 3.7 3.8

Increasing and Decreasing Functions Extrema and the First-Derivative Test Concavity and the Second-Derivative Test Optimization Problems Business and Economics Applications Asymptotes Curve Sketching: A Summary Differentials and Marginal Analysis 205

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206

CHAPTER 3

Applications of the Derivative

Section 3.1 ■ Test for increasing and decreasing functions.

Increasing and Decreasing Functions

■ Find the critical numbers of functions and find the open intervals on

which functions are increasing or decreasing. ■ Use increasing and decreasing functions to model and solve real-life

problems.

Increasing and Decreasing Functions A function is increasing if its graph moves up as x moves to the right and decreasing if its graph moves down as x moves to the right. The following definition states this more formally. Definition of Increasing and Decreasing Functions

A function f is increasing on an interval if for any two numbers x1 and x2 in the interval y

x=a

x2 > x1

x=b

A function f is decreasing on an interval if for any two numbers x1 and x2 in the interval implies f 共x2兲 < f 共x1兲.

ng

Inc

asi

rea

cre

sing

De

x2 > x1

Constant

f (x)

implies f 共x2兲 > f 共x1兲.

0

FIGURE 3.1

f (x)

0 f (x)

0

x

The function in Figure 3.1 is decreasing on the interval 共⫺ ⬁, a兲, constant on the interval 共a, b兲, and increasing on the interval 共b, ⬁兲. Actually, from the definition of increasing and decreasing functions, the function shown in Figure 3.1 is decreasing on the interval 共⫺ ⬁, a兴 and increasing on the interval 关b, ⬁兲. This text restricts the discussion to finding open intervals on which a function is increasing or decreasing. The derivative of a function can be used to determine whether the function is increasing or decreasing on an interval. Test for Increasing and Decreasing Functions

Let f be differentiable on the interval 共a, b兲. 1. If f⬘共x兲 > 0 for all x in 共a, b兲, then f is increasing on 共a, b兲. 2. If f⬘共x兲 < 0 for all x in 共a, b兲, then f is decreasing on 共a, b兲. 3. If f⬘共x兲 ⫽ 0 for all x in 共a, b兲, then f is constant on 共a, b兲.

STUDY TIP The conclusions in the first two cases of testing for increasing and decreasing functions are valid even if f ⬘共x兲 ⫽ 0 at a finite number of x-values in 共a, b兲.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 3.1 y

Example 1

4

is decreasing on the open interval 共⫺ ⬁, 0兲 and increasing on the open interval 共0, ⬁兲.

2

f (x)

−1

1

, 0)

(0,

Decreasing

0 2

x

)

8

(

SOLUTION

1

0

8

−2

Testing for Increasing and Decreasing Functions

f 共x兲 ⫽ x 2

3

f (x)

207

Show that the function

x2

f (x)

Increasing and Decreasing Functions

Increasing

FIGURE 3.2

The derivative of f is

f⬘共x兲 ⫽ 2x. On the open interval 共⫺ ⬁, 0兲, the fact that x is negative implies that f⬘共x兲 ⫽ 2x is also negative. So, by the test for a decreasing function, you can conclude that f is decreasing on this interval. Similarly, on the open interval 共0, ⬁兲, the fact that x is positive implies that f⬘共x兲 ⫽ 2x is also positive. So, it follows that f is increasing on this interval, as shown in Figure 3.2.

✓CHECKPOINT 1 Use a graphing utility to graph f 共x兲 ⫽ 2 ⫺ x 2 and f⬘共x兲 ⫽ ⫺2x in the same viewing window. On what interval is f increasing? On what interval is f⬘ positive? Describe how the first derivative can be used to determine where a function is increasing and decreasing. Repeat this analysis for g共x兲 ⫽ x3 ⫺ x and g⬘共x兲 ⫽ 3x 2 ⫺ 1.

✓CHECKPOINT 2 From 1995 through 2004, the consumption W of bottled water in the United States (in gallons per person per year) can be modeled by W ⫽ 0.058t 2 ⫹ 0.19t ⫹ 9.2, 5 ≤ t ≤ 14 where t ⫽ 5 corresponds to 1995. Show that the consumption of bottled water was increasing from 1995 to 2004. (Source: U.S. Department of Agriculture) ■

Show that the function f 共x兲 ⫽ x 4 is decreasing on the open interval 共⫺ ⬁, 0兲 and increasing on the open interval 共0, ⬁兲. ■

Example 2

Modeling Consumption

From 1997 through 2004, the consumption C of Italian cheeses in the United States (in pounds per person per year) can be modeled by C ⫽ ⫺0.0333t2 ⫹ 0.996t ⫹ 5.40,

7 ≤ t ≤ 14

where t ⫽ 7 corresponds to 1997 (see Figure 3.3). Show that the consumption of Italian cheeses was increasing from 1997 to 2004. (Source: U.S. Department of Agriculture)

The derivative of this model is dC兾dt ⫽ ⫺0.0666t ⫹ 0.996. For the open interval 共7, 14兲, the derivative is positive. So, the function is increasing, which implies that the consumption of Italian cheeses was increasing during the given time period.

SOLUTION

Italian Cheese Consumption C

Consumption (in pounds per person)

D I S C O V E RY

13.0 12.5 12.0 11.5 11.0 10.5 10.0 7

8

9

10

11

12

13

14

t

Year (7 ↔ 1997)

FIGURE 3.3

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CHAPTER 3

Applications of the Derivative

Critical Numbers and Their Use In Example 1, you were given two intervals: one on which the function was decreasing and one on which it was increasing. Suppose you had been asked to determine these intervals. To do this, you could have used the fact that for a continuous function, f⬘共x兲 can change signs only at x-values where f⬘共x兲 ⫽ 0 or at x-values where f⬘共x兲 is undefined, as shown in Figure 3.4. These two types of numbers are called the critical numbers of f. y

y

0

f (x)

f (x)

0

f (x)

0

g in as

0

sin g

re

g

f (c)

rea

re

c De

in as cre

as in

g

De x

c

0

In c

f (x)

In c

208

c

x

f (c) is undefined.

FIGURE 3.4

Definition of Critical Number

If f is defined at c, then c is a critical number of f if f⬘共c兲 ⫽ 0 or if f⬘ 共c兲 is undefined.

STUDY TIP This definition requires that a critical number be in the domain of the function. For example, x ⫽ 0 is not a critical number of the function f 共x兲 ⫽ 1兾x.

To determine the intervals on which a continuous function is increasing or decreasing, you can use the guidelines below. Guidelines for Applying Increasing/Decreasing Test

1. Find the derivative of f. 2. Locate the critical numbers of f and use these numbers to determine test intervals. That is, find all x for which f⬘共x兲 ⫽ 0 or f⬘共x兲 is undefined. 3. Test the sign of f⬘共x兲 at an arbitrary number in each of the test intervals. 4. Use the test for increasing and decreasing functions to decide whether f is increasing or decreasing on each interval.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 3.1

Example 3

Increasing and Decreasing Functions

209

Finding Increasing and Decreasing Intervals

Find the open intervals on which the function is increasing or decreasing. 3 f 共x兲 ⫽ x3 ⫺ x 2 2 Begin by finding the derivative of f. Then set the derivative equal to zero and solve for the critical numbers. SOLUTION

y

In c re a

sing

2

1

(0, 0)

x

De

1

cre

g

Incr

easin

−1

f⬘共x兲 ⫽ 3x 2 ⫺ 3x 3x 2 ⫺ 3x ⫽ 0 3共x兲共x ⫺ 1兲 ⫽ 0 x ⫽ 0, x ⫽ 1

3 2 x 2

x3

f (x)

−1

asi

Differentiate original function. Set derivative equal to 0. Factor. Critical numbers

Because there are no x-values for which f⬘ is undefined, it follows that x ⫽ 0 and x ⫽ 1 are the only critical numbers. So, the intervals that need to be tested are 共⫺ ⬁, 0兲, 共0, 1兲, and 共1, ⬁兲. The table summarizes the testing of these three intervals.

2

(

ng 1, − 12

(

FIGURE 3.5

✓CHECKPOINT 3 Find the open intervals on which the function f 共x兲 ⫽ x3 ⫺ 12x is increasing or decreasing. ■

Interval

⫺⬁ < x < 0

0 < x < 1 1 2

1 < x
0

f⬘ 共12 兲 ⫽ ⫺ 34 < 0

f⬘ 共2兲 ⫽ 6 > 0

Conclusion

Increasing

Decreasing

Increasing

x⫽2

The graph of f is shown in Figure 3.5. Note that the test values in the intervals were chosen for convenience—other x-values could have been used.

TECHNOLOGY You can use the trace feature of a graphing utility to confirm the result of Example 3. Begin by graphing the function, as shown at the right. Then activate the trace feature and move the cursor from left to right. In intervals on which the function is increasing, note that the y-values increase as the x-values increase, whereas in intervals on which the function is decreasing, the y-values decrease as the x-values increase.*

4

3

−1

−2

On this interval, the y-values increase as the x-values increase.

On this interval, the y-values decrease as the x-values increase.

On this interval, the y-values increase as the x-values increase.

*Specific calculator keystroke instructions for operations in this and other technology boxes can be found at college.hmco.com/info/larsonapplied.

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210

CHAPTER 3

Applications of the Derivative

Not only is the function in Example 3 continuous on the entire real line, it is also differentiable there. For such functions, the only critical numbers are those for which f⬘共x兲 ⫽ 0. The next example considers a continuous function that has both types of critical numbers—those for which f⬘共x兲 ⫽ 0 and those for which f⬘ 共x兲 is undefined.

Example 4

Algebra Review For help on the algebra in Example 4, see Example 2(d) in the Chapter 3 Algebra Review, on page 284.

Finding Increasing and Decreasing Intervals

Find the open intervals on which the function f 共x兲 ⫽ 共x 2 ⫺ 4兲2兾3 is increasing or decreasing. SOLUTION

Begin by finding the derivative of the function.

2 f⬘共x兲 ⫽ 共x 2 ⫺ 4兲⫺1兾3共2x兲 3 ⫽

Differentiate.

4x 3共x 2 ⫺ 4兲1兾3

Simplify.

From this, you can see that the derivative is zero when x ⫽ 0 and the derivative is undefined when x ⫽ ± 2. So, the critical numbers are x ⫽ ⫺2, x ⫽ 0,

and x ⫽ 2.

Critical numbers

This implies that the test intervals are y

(x 2

f (x)

4)

2 3

g sin rea

Inc

ng ng

asi

cre

De

sing (−2, 0)

easi

rea −4 −3 −2 −1

2

( 0, 2 3 2 )

Incr

Dec

4

Test intervals

The table summarizes the testing of these four intervals, and the graph of the function is shown in Figure 3.6.

6 5

共⫺ ⬁, ⫺2兲, 共⫺2, 0兲, 共0, 2兲, and 共2, ⬁兲.

1 1

2

3

4

(2, 0)

x

Interval

⫺ ⬁ < x < ⫺2

⫺2 < x < 0

0 < x < 2

2 < x
0

f⬘ 共1兲 < 0

f⬘ 共3兲 > 0

Conclusion

Decreasing

Increasing

Decreasing

Increasing



FIGURE 3.6

✓CHECKPOINT 4 Find the open intervals on which the function f 共x兲 ⫽ x2兾3 is increasing or decreasing. ■ STUDY TIP To test the intervals in the table, it is not necessary to evaluate f⬘共x兲 at each test value—you only need to determine its sign. For example, you can determine the sign of f⬘共⫺3兲 as shown. f⬘共⫺3兲 ⫽

4共⫺3兲 negative ⫽ ⫽ negative 3共9 ⫺ 4兲1兾3 positive

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SECTION 3.1

Increasing and Decreasing Functions

211

The functions in Examples 1 through 4 are continuous on the entire real line. If there are isolated x-values at which a function is not continuous, then these x-values should be used along with the critical numbers to determine the test intervals. For example, the function f 共x兲 ⫽

x4 ⫹ 1 x2

is not continuous when x ⫽ 0. Because the derivative of f

1

x2

g asin

x ⫽ ⫺1, x ⫽ 1 x⫽0

3

(−1, 2) −3

−2

2 1

(1, 2)

−1

2共x 4 ⫺ 1兲 x3

is zero when x ⫽ ± 1, you should use the following numbers to determine the test intervals.

Incr e

Increasing

ng

Decreasing

i reas Dec

4

x4

f (x)

y

f⬘共x兲 ⫽

1

2

x

3

FIGURE 3.7

Critical numbers Discontinuity

After testing f⬘共x兲, you can determine that the function is decreasing on the intervals 共⫺ ⬁, ⫺1兲 and 共0, 1兲, and increasing on the intervals 共⫺1, 0兲 and 共1, ⬁兲, as shown in Figure 3.7. The converse of the test for increasing and decreasing functions is not true. For instance, it is possible for a function to be increasing on an interval even though its derivative is not positive at every point in the interval.

Example 5

Testing an Increasing Function

Show that f 共x兲 ⫽ x3 ⫺ 3x 2 ⫹ 3x is increasing on the entire real line. SOLUTION

From the derivative of f

f⬘共x兲 ⫽ 3x 2 ⫺ 6x ⫹ 3 ⫽ 3共x ⫺ 1兲2 y

f (x)

x3

3x 2

3x

you can see that the only critical number is x ⫽ 1. So, the test intervals are 共⫺ ⬁, 1兲 and 共1, ⬁兲. The table summarizes the testing of these two intervals. From Figure 3.8, you can see that f is increasing on the entire real line, even though f⬘共1兲 ⫽ 0. To convince yourself of this, look back at the definition of an increasing function.

2

1

−1

(1, 1)

1

2

x

Interval

⫺⬁ < x < 1

1 < x
0

f⬘ 共2兲 ⫽ 3共1兲2 > 0

Conclusion

Increasing

Increasing



FIGURE 3.8

✓CHECKPOINT 5 Show that f 共x兲 ⫽ ⫺x3 ⫹ 2 is decreasing on the entire real line.



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212

CHAPTER 3

Applications of the Derivative

Application Example 6

Profit Analysis

A national toy distributor determines the cost and revenue models for one of its games. C ⫽ 2.4x ⫺ 0.0002x 2, 0 ≤ x ≤ 6000 R ⫽ 7.2x ⫺ 0.001x 2, 0 ≤ x ≤ 6000 Determine the interval on which the profit function is increasing. SOLUTION

P⫽R⫺C ⫽ 共7.2x ⫺ 0.001x 2兲 ⫺ 共2.4x ⫺ 0.0002x 2兲 ⫽ 4.8x ⫺ 0.0008x 2.

Revenue, cost, and profit (in dollars)

Profit Analysis

To find the interval on which the profit is increasing, set the marginal profit P⬘ equal to zero and solve for x.

Revenue

12,000

P⬘ ⫽ 4.8 ⫺ 0.0016x 4.8 ⫺ 0.0016x ⫽ 0 ⫺0.0016x ⫽ ⫺4.8

10,000 8,000

(3000, 7200)

6,000 4,000 2,000

Cost Profit 2,000

4,000

6,000

Number of games

FIGURE 3.9

The profit for producing x games is

x

⫺4.8 x⫽ ⫺0.0016 x ⫽ 3000 games

Differentiate profit function. Set P⬘ equal to 0. Subtract 4.8 from each side. Divide each side by ⫺0.0016. Simplify.

On the interval 共0, 3000兲, P⬘ is positive and the profit is increasing. On the interval 共3000, 6000兲, P⬘ is negative and the profit is decreasing. The graphs of the cost, revenue, and profit functions are shown in Figure 3.9.

✓CHECKPOINT 6 A national distributor of pet toys determines the cost and revenue functions for one of its toys. C ⫽ 1.2x ⫺ 0.0001x2,

0 ≤ x ≤ 6000

R ⫽ 3.6x ⫺ 0.0005x2,

0 ≤ x ≤ 6000

Determine the interval on which the profit function is increasing.



CONCEPT CHECK 1. Write a verbal description of (a) the graph of an increasing function and (b) the graph of a decreasing function. 2. Complete the following: If f⬘ 冇x冈 > 0 for all x in 冇a, b冈, then f is ______ on 冇a, b冈. [Assume f is differentiable on 冇a, b冈.] 3. If f is defined at c, under what condition(s) is c a critical number of f ? 4. In your own words, state the guidelines for determining the intervals on which a continuous function is increasing or decreasing.

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SECTION 3.1

Skills Review 3.1

213

Increasing and Decreasing Functions

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 0.3 and 1.4.

In Exercises 1– 4, solve the equation. 5 2. 15x ⫽ x 2 8

1. x 2 ⫽ 8x 3.

x 2 ⫺ 25 ⫽0 x3

4.

2x 冪1 ⫺ x 2

⫽0

In Exercises 5– 8, find the domain of the expression. 5.

x⫹3 x⫺3

6.

7.

2x ⫹ 1 x 2 ⫺ 3x ⫺ 10

8.

2 冪1 ⫺ x

3x 冪9 ⫺ 3x 2

In Exercises 9–12, evaluate the expression when x ⴝ ⴚ2, 0, and 2. 9. ⫺2共x ⫹ 1兲共x ⫺ 1兲 11.

10. 4共2x ⫹ 1兲共2x ⫺ 1兲

2x ⫹ 1 共x ⫺ 1兲2

12.

Exercises 3.1

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–4, evaluate the derivative of the function at the indicated points on the graph. x2 x2 ⫹ 4

1. f 共x兲 ⫽

2. f 共x兲 ⫽ x ⫹

6

x

(0, 0) 1

−1

4

−3

−2

−1

1

6

2

x

−3

x

8 10

(

2 2 3 − , 3 3

(0, 0)

2 −1

1

(−2, 0)

1

4 3

−2 −1 −2 −3 −4

−4

y

3

−4

y

4. f 共x兲 ⫽ ⫺3x冪x ⫹ 1

4

(−3, 1)

(−1, 0)

x

x3 ⫺ 3x 4

−2

y

(−1, 1)

−1

2 2

3. f 共x兲 ⫽ 共x ⫹ 2兲2兾3

−4 −3

(4, 6)

4

6. f 共x兲 ⫽

y

(8, 172 )

8

(1, 15 )

In Exercises 5 – 8, use the derivative to identify the open intervals on which the function is increasing or decreasing. Verify your result with the graph of the function. 5. f 共x兲 ⫽ ⫺ 共x ⫹ 1兲2

(2, 10)

10 2

(− 1, 15 )

32 x2

y

y

1

⫺2共x ⫹ 1兲 共x ⫺ 4兲2

1

)

7. f 共x兲 ⫽ x 4 ⫺ 2x 2

8. f 共x兲 ⫽

1 2

4

x

x2 x⫹1 y

y 3

x

2 1

−1 −3 −2

2

3

x

−6 −4 −2 −2

2

4

6

−2 −3

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

x

CHAPTER 3

Applications of the Derivative

In Exercises 9–32, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. 9. f 共x兲 ⫽ 2x ⫺ 3

10. f 共x兲 ⫽ 5 ⫺ 3x

11. g共x兲 ⫽ ⫺ 共x ⫺ 1兲

12. g共x兲 ⫽ 共x ⫹ 2兲2

13. y ⫽ x 2 ⫺ 6x

14. y ⫽ ⫺x 2 ⫹ 2x

15. y ⫽ x3 ⫺ 6x 2

16. y ⫽ 共x ⫺ 2兲3

17. f 共x兲 ⫽ 冪x 2 ⫺ 1

18. f 共x兲 ⫽ 冪9 ⫺ x 2

19. y ⫽ x1兾3 ⫹ 1

2兾3 20. y ⫽ x ⫺ 4 22. g共x兲 ⫽ 共x ⫺ 1兲2兾3

2

1兾3 21. g共x兲 ⫽ 共x ⫺ 1兲 23. f 共x兲 ⫽ ⫺2x 2 ⫹ 4x ⫹ 3

25. y ⫽

3x3



12x 2

⫹ 15x

24. f 共x兲 ⫽ x 2 ⫹ 8x ⫹ 10 3 x ⫺ 1 28. h共x兲 ⫽ x 冪

29. f 共x兲 ⫽ x 4 ⫺ 2x3

1 30. f 共x兲 ⫽ 4x 4 ⫺ 2x 2

x x2 ⫹ 4

32. f 共x兲 ⫽

x2 x2 ⫹ 4

x 34. f 共x兲 ⫽ x⫹1

⫺x , x ≤ 0 冦4⫺2x, x > 0 3x ⫹ 1, x ≤ 1 37. y ⫽ 冦 5⫺x, x > 1 ⫺x ⫹ 1, x ≤ 0 38. y ⫽ 冦 ⫺x ⫹ 2x, x > 0 2

35. y ⫽

36. y ⫽

冦2xx ⫺⫹ 2,1, 2

x ≤ ⫺1 x > ⫺1

2

3

2

39. Cost The ordering and transportation cost C (in hundreds of dollars) for an automobile dealership is modeled by 1 x C ⫽ 10 ⫹ , x ≥ 1 x x⫹3



273 K 1273 K 2273 K 1000

2000

3000

Velocity (in meters per second)

41. Medical Degrees The number y of medical degrees conferred in the United States from 1970 through 2004 can be modeled by y ⫽ 0.813t3 ⫺ 55.70t2 ⫹ 1185.2t ⫹ 7752, 0 ≤ t ≤ 34

In Exercises 33–38, find the critical numbers and the open intervals on which the function is increasing or decreasing. (Hint: Check for discontinuities.) Sketch the graph of the function. 2x 33. f 共x兲 ⫽ 16 ⫺ x 2

Molecular Velocity

26. y ⫽ x 3 ⫺ 3x ⫹ 2

27. f 共x兲 ⫽ x冪x ⫹ 1

31. f 共x兲 ⫽

average velocities (indicated by the peaks of the curves) for the three temperatures, and describe the intervals on which the velocity is increasing and decreasing for each of the three temperatures. (Source: Adapted from Zumdahl, Chemistry, Seventh Edition)

Number of N2 (nitrogen) molecules

214



where x is the number of automobiles ordered. (a) Find the intervals on which C is increasing or decreasing. (b) Use a graphing utility to graph the cost function. (c) Use the trace feature to determine the order sizes for which the cost is $900. Assuming that the revenue function is increasing for x ≥ 0, which order size would you use? Explain your reasoning. 40. Chemistry: Molecular Velocity Plots of the relative numbers of N2 (nitrogen) molecules that have a given velocity at each of three temperatures (in degrees Kelvin) are shown in the figure. Identify the differences in the

where t is the time in years, with t ⫽ 0 corresponding to 1970. (Source: U.S. National Center for Education Statistics) (a) Use a graphing utility to graph the model. Then graphically estimate the years during which the model is increasing and the years during which it is decreasing. (b) Use the test for increasing and decreasing functions to verify the result of part (a). 42. MAKE A DECISION: PROFIT The profit P made by a cinema from selling x bags of popcorn can be modeled by P ⫽ 2.36x ⫺

x2 ⫺ 3500, 0 ≤ x ≤ 50,000. 25,000

(a) Find the intervals on which P is increasing and decreasing. (b) If you owned the cinema, what price would you charge to obtain a maximum profit for popcorn? Explain your reasoning. 43. Profit Analysis A fast-food restaurant determines the cost and revenue models for its hamburgers. C ⫽ 0.6x ⫹ 7500, 0 ≤ x ≤ 50,000 R⫽

1 共65,000x ⫺ x2兲, 0 ≤ x ≤ 50,000 20,000

(a) Write the profit function for this situation. (b) Determine the intervals on which the profit function is increasing and decreasing. (c) Determine how many hamburgers the restaurant needs to sell to obtain a maximum profit. Explain your reasoning.

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SECTION 3.2

215

Extrema and the First-Derivative Test

Section 3.2 ■ Recognize the occurrence of relative extrema of functions.

Extrema and the First-Derivative Test

■ Use the First-Derivative Test to find the relative extrema of functions. ■ Find absolute extrema of continuous functions on a closed interval. ■ Find minimum and maximum values of real-life models and interpret the

results in context.

Relative Extrema y

You have used the derivative to determine the intervals on which a function is increasing or decreasing. In this section, you will examine the points at which a function changes from increasing to decreasing, or vice versa. At such a point, the function has a relative extremum. (The plural of extremum is extrema.) The relative extrema of a function include the relative minima and relative maxima of the function. For instance, the function shown in Figure 3.10 has a relative maximum at the left point and a relative minimum at the right point.

ing as cre

sin cr

In

g

In

sin

ea

cr

ea

De

g

Relative maximum

Relative minimum x

FIGURE 3.10

Definition of Relative Extrema

Let f be a function defined at c. 1. f 共c兲 is a relative maximum of f if there exists an interval 共a, b兲 containing c such that f 共x兲 ≤ f 共c兲 for all x in 共a, b兲. 2. f 共c兲 is a relative minimum of f if there exists an interval 共a, b兲 containing c such that f 共x兲 ≥ f 共c兲 for all x in 共a, b兲. If f 共c兲 is a relative extremum of f, then the relative extremum is said to occur at x ⫽ c. For a continuous function, the relative extrema must occur at critical numbers of the function, as shown in Figure 3.11. y

y

Relative maximum f ′(c) = 0

c

Relative maximum f ′(c) is undefined.

Horizontal tangent

x

c

x

FIGURE 3.11

Occurrences of Relative Extrema

If f has a relative minimum or relative maximum when x ⫽ c, then c is a critical number of f. That is, either f⬘共c兲 ⫽ 0 or f⬘共c兲 is undefined.

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216

CHAPTER 3

Applications of the Derivative

The First-Derivative Test D I S C O V E RY Use a graphing utility to graph the function f 共x兲 ⫽ x 2 and its first derivative f⬘ 共x兲 ⫽ 2x in the same viewing window. Where does f have a relative minimum? What is the sign of f⬘ to the left of this relative minimum? What is the sign of f⬘ to the right? Describe how the sign of f⬘ can be used to determine the relative extrema of a function.

The discussion on the preceding page implies that in your search for relative extrema of a continuous function, you only need to test the critical numbers of the function. Once you have determined that c is a critical number of a function f, the First-Derivative Test for relative extrema enables you to classify f 共c兲 as a relative minimum, a relative maximum, or neither. First-Derivative Test for Relative Extrema

Let f be continuous on the interval 共a, b兲 in which c is the only critical number. If f is differentiable on the interval (except possibly at c), then f 共c兲 can be classified as a relative minimum, a relative maximum, or neither, as shown. 1. On the interval 共a, b兲, if f⬘共x兲 is negative to the left of x ⫽ c and positive to the right of x ⫽ c, then f 共c兲 is a relative minimum. 2. On the interval 共a, b兲, if f⬘共x兲 is positive to the left of x ⫽ c and negative to the right of x ⫽ c, then f 共c兲 is a relative maximum. 3. On the interval 共a, b兲, if f⬘共x兲 is positive on both sides of x ⫽ c or negative on both sides of x ⫽ c, then f 共c兲 is not a relative extremum of f. A graphical interpretation of the First-Derivative Test is shown in Figure 3.12. c f ′(x) is positive.

Relative minimum f ′(x) is negative.

c

Relative maximum f ′(x) is positive.

f ′(x) is positive. f ′(x) is positive.

f ′(x) is negative.

c Neither minimum nor maximum

f ′(x) is negative. c

f ′(x) is negative.

Neither minimum nor maximum

FIGURE 3.12

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SECTION 3.2

Example 1

Extrema and the First-Derivative Test

217

Finding Relative Extrema

Find all relative extrema of the function f 共x兲 ⫽ 2x3 ⫺ 3x 2 ⫺ 36x ⫹ 14. Begin by finding the critical numbers of f.

SOLUTION

f⬘共x兲 ⫽ 6x 2 ⫺ 6x ⫺ 36 6x 2 ⫺ 6x ⫺ 36 ⫽ 0 6共x 2 ⫺ x ⫺ 6兲 ⫽ 0 6共x ⫺ 3兲共x ⫹ 2兲 ⫽ 0 x ⫽ ⫺2, x ⫽ 3

Find derivative of f. Set derivative equal to 0. Factor out common factor. Factor. Critical numbers

Because f⬘共x兲 is defined for all x, the only critical numbers of f are x ⫽ ⫺2 and x ⫽ 3. Using these numbers, you can form the three test intervals 共⫺ ⬁, ⫺2兲, 共⫺2, 3兲, and 共3, ⬁兲. The testing of the three intervals is shown in the table. Interval

⫺ ⬁ < x < ⫺2

⫺2 < x < 3

3 < x
0

f⬘ 共0兲 ⫽ ⫺36 < 0

f⬘ 共4兲 ⫽ 36 > 0

Conclusion

Increasing

Decreasing

Increasing



Using the First-Derivative Test, you can conclude that the critical number ⫺2 yields a relative maximum 关 f⬘共x兲 changes sign from positive to negative兴, and the critical number 3 yields a relative minimum 关 f⬘共x兲 changes sign from negative to positive兴. Relative maximum (−2, 58)

STUDY TIP In Section 2.2, Example 8, you examined the graph of the function f 共x兲 ⫽ x 3 ⫺ 4x ⫹ 2 and discovered that it does not have a relative minimum at the point 共1, ⫺1兲. Try using the First-Derivative Test to find the point at which the graph does have a relative minimum.

y

f(x) = 2x 3 − 3x 2 − 36x + 14

75

25 −3 −2 −1

2 3 4

−50 −75

(3, − 67)

x

Relative minimum

FIGURE 3.13

The graph of f is shown in Figure 3.13. The relative maximum is f 共⫺2兲 ⫽ 58 and the relative minimum is f 共3兲 ⫽ ⫺67.

✓CHECKPOINT 1 Find all relative extrema of f 共x兲 ⫽ 2x3 ⫺ 6x ⫹ 1.



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218

CHAPTER 3

Applications of the Derivative

In Example 1, both critical numbers yielded relative extrema. In the next example, only one of the two critical numbers yields a relative extremum.

Example 2

Algebra Review For help on the algebra in Example 2, see Example 2(c) in the Chapter 3 Algebra Review, on page 284.

Finding Relative Extrema

Find all relative extrema of the function f 共x兲 ⫽ x 4 ⫺ x 3. SOLUTION

From the derivative of the function

f⬘共x兲 ⫽ 4x3 ⫺ 3x2 ⫽ x2共4x ⫺ 3兲 you can see that the function has only two critical numbers: x ⫽ 0 and x ⫽ 34. These numbers produce the test intervals 共⫺ ⬁, 0兲, 共0, 34 兲, and 共34, ⬁兲, which are tested in the table.

y f ( x)

x4

x3

1

(0, 0)

−1

(

3 , 4

27 − 256

)

⫺⬁ < x < 0

Interval

0 < x
1. When the price is $10 per unit, the quantity demanded is eight units. The initial cost is $100 and the cost per unit is $4. What price will yield a maximum profit?

49. Profit When soft drinks were sold for $1.00 per can at football games, approximately 6000 cans were sold. When the price was raised to $1.20 per can, the quantity demanded dropped to 5600. The initial cost is $5000 and the cost per unit is $0.50. Assuming that the demand function is linear, use the table feature of a graphing utility to determine the price that will yield a maximum profit. 50. Medical Science Coughing forces the trachea (windpipe) to contract, which in turn affects the velocity of the air through the trachea. The velocity of the air during coughing can be modeled by v ⫽ k共R ⫺ r兲r 2, 0 ≤ r < R, where k is a constant, R is the normal radius of the trachea, and r is the radius during coughing. What radius r will produce the maximum air velocity? 51. Population The resident population P (in millions) of the United States from 1790 through 2000 can be modeled by P ⫽ 0.00000583t3 ⫹ 0.005003t2 ⫹ 0.13776t ⫹ 4.658, ⫺10 ≤ t ≤ 200, where t ⫽ 0 corresponds to 1800. (Source: U.S. Census Bureau) (a) Make a conjecture about the maximum and minimum populations in the U.S. from 1790 to 2000. (b) Analytically find the maximum and minimum populations over the interval. (c) Write a brief paragraph comparing your conjecture with your results in part (b). 52. Biology: Fertility Rates The graph of the United States fertility rate shows the number of births per 1000 women in their lifetime according to the birth rate in that particular year. (Source: U.S. National Center for Health Statistics) (a) Around what year was the fertility rate the highest, and to how many births per 1000 women did this rate correspond? (b) During which time periods was the fertility rate increasing most rapidly? Most slowly? (c) During which time periods was the fertility rate decreasing most rapidly? Most slowly? (d) Give some possible real-life reasons for fluctuations in the fertility rate. United States Fertility

y

Fertility rate (in births per 1000 women)

224

2500 2400 2300 2200 2100 2000 1900 1800 1700 3

6

9

12 15 18 21 24 27 30 33

t

Year (0 ↔ 1970)

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SECTION 3.3

Concavity and the Second-Derivative Test

225

Section 3.3

Concavity and the Second-Derivative Test

■ Determine the intervals on which the graphs of functions are concave

upward or concave downward. ■ Find the points of inflection of the graphs of functions. ■ Use the Second-Derivative Test to find the relative extrema of functions. ■ Find the points of diminishing returns of input-output models.

Concavity You already know that locating the intervals over which a function f increases or decreases is helpful in determining its graph. In this section, you will see that locating the intervals on which f⬘ increases or decreases can determine where the graph of f is curving upward or curving downward. This property of curving upward or downward is defined formally as the concavity of the graph of the function.

y

Concave upward, f is increasing.

Definition of Concavity

Let f be differentiable on an open interval I. The graph of f is x

1. concave upward on I if f⬘ is increasing on the interval. 2. concave downward on I if f⬘ is decreasing on the interval.

y

From Figure 3.20, you can observe the following graphical interpretation of concavity. 1. A curve that is concave upward lies above its tangent line. Concave downward, f is decreasing.

2. A curve that is concave downward lies below its tangent line.

x

FIGURE 3.20

This visual test for concavity is useful when the graph of a function is given. To determine concavity without seeing a graph, you need an analytic test. It turns out that you can use the second derivative to determine these intervals in much the same way that you use the first derivative to determine the intervals on which f is increasing or decreasing. Test for Concavity

Let f be a function whose second derivative exists on an open interval I. 1. If f ⬙ 共x兲 > 0 for all x in I, then f is concave upward on I. 2. If f ⬙ 共x兲 < 0 for all x in I, then f is concave downward on I.

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226

CHAPTER 3

Applications of the Derivative

For a continuous function f, you can find the open intervals on which the graph of f is concave upward and concave downward as follows. [For a function that is not continuous, the test intervals should be formed using points of discontinuity, along with the points at which f ⬙ 共x兲 is zero or undefined.] D I S C O V E RY

Guidelines for Applying Concavity Test

Use a graphing utility to graph the function f 共x兲 ⫽ x3 ⫺ x and its second derivative f ⬙ 共x兲 ⫽ 6x in the same viewing window. On what interval is f concave upward? On what interval is f ⬙ positive? Describe how the second derivative can be used to determine where a function is concave upward and concave downward. Repeat this analysis for the functions g共x兲 ⫽ x 4 ⫺ 6x2 and g⬙ 共x兲 ⫽ 12x2 ⫺ 12.

1. Locate the x-values at which f ⬙ 共x兲 ⫽ 0 or f ⬙ 共x兲 is undefined. 2. Use these x-values to determine the test intervals. 3. Test the sign of f ⬙ 共x兲 in each test interval.

Example 1

Applying the Test for Concavity

a. The graph of the function f 共x兲 ⫽ x2

Original function

is concave upward on the entire real line because its second derivative f ⬙ 共x兲 ⫽ 2

Second derivative

is positive for all x. (See Figure 3.21.) b. The graph of the function f 共x兲 ⫽ 冪x

Original function

is concave downward for x > 0 because its second derivative 1 f ⬙ 共x兲 ⫽ ⫺ x⫺3兾2 4

Second derivative

is negative for all x > 0. (See Figure 3.22.) y

y

✓CHECKPOINT 1 a. Find the second derivative of f 共x兲 ⫽ ⫺2x2 and discuss the concavity of the graph. b. Find the second derivative of f 共x兲 ⫽ ⫺2冪x and discuss the concavity of the graph. ■

4

4

3

3

2

2

f(x) = x 2

1

−2

−1

FIGURE 3.21

f(x) =

1

x

x

1

2

Concave Upward

1

FIGURE 3.22

2

3

4

x

Concave Downward

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SECTION 3.3

Example 2 Algebra Review For help on the algebra in Example 2, see Example 1(a) in the Chapter 3 Algebra Review, on page 283.

Determining Concavity

Determine the open intervals on which the graph of the function is concave upward or concave downward. f 共x兲 ⫽

6 x2 ⫹ 3 Begin by finding the second derivative of f.

SOLUTION

f 共x兲 ⫽ 6共x2 ⫹ 3兲⫺1 f⬘共x兲 ⫽ 共⫺6兲共2x兲共x2 ⫹ 3兲⫺2

Rewrite original function. Chain Rule

⫺12x ⫽ 2 共x ⫹ 3兲2 共x2 ⫹ 3兲2共⫺12兲 ⫺ 共⫺12x兲共2兲共2x兲共x2 ⫹ 3兲 f ⬙ 共x兲 ⫽ 共x2 ⫹ 3兲4 ⫺12共x2 ⫹ 3兲 ⫹ 共48x2兲 ⫽ 共x2 ⫹ 3兲3 36共x2 ⫺ 1兲 ⫽ 2 共x ⫹ 3兲3

STUDY TIP In Example 2, f⬘ is increasing on the interval 共1, ⬁兲 even though f is decreasing there. Be sure you see that the increasing or decreasing of f⬘ does not necessarily correspond to the increasing or decreasing of f.

227

Concavity and the Second-Derivative Test

Simplify. Quotient Rule Simplify. Simplify.

From this, you can see that f ⬙ 共x兲 is defined for all real numbers and f ⬙ 共x兲 ⫽ 0 when x ⫽ ± 1. So, you can test the concavity of f by testing the intervals 共⫺ ⬁, ⫺1兲, 共⫺1, 1兲, and 共1, ⬁兲, as shown in the table. The graph of f is shown in Figure 3.23. Interval

⫺ ⬁ < x < ⫺1

⫺1 < x < 1

1 < x
0

f ⬙ 共0兲 < 0

f ⬙ 共2兲 > 0

Conclusion

Concave upward

Concave downward

Concave upward



y

4

6

f )x)

✓CHECKPOINT 2 Determine the intervals on which the graph of the function is concave upward or concave downward. f 共x兲 ⫽

12 x2 ⫹ 4



x2

3 3

Concave upward, 0 f ″)x)

−3

−2

−1

Concave downward, f ″)x) 0 Concave upward, f ″)x) 0

1

1

2

3

x

FIGURE 3.23

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228

CHAPTER 3

Applications of the Derivative

Points of Inflection If the tangent line to a graph exists at a point at which the concavity changes, then the point is a point of inflection. Three examples of inflection points are shown in Figure 3.24. (Note that the third graph has a vertical tangent line at its point of inflection.) STUDY TIP As shown in Figure 3.24, a graph crosses its tangent line at a point of inflection.

y

y

Point of inflection

Concave downward

Concave upward

FIGURE 3.24

y

Concave upward Point of inflection Concave downward

x

Concave downward Concave Point of upward inflection x

x

The graph crosses its tangent line at a point of inflection.

Definition of Point of Inflection

If the graph of a continuous function has a tangent line at a point where its concavity changes from upward to downward (or downward to upward), then the point is a point of inflection.

D I S C O V E RY Use a graphing utility to graph f 共x兲 ⫽ x3 ⫺ 6x2 ⫹ 12x ⫺ 6

and

f ⬙ 共x兲 ⫽ 6x ⫺ 12

in the same viewing window. At what x-value does f ⬙ 共x兲 ⫽ 0? At what x-value does the point of inflection occur? Repeat this analysis for g共x兲 ⫽ x 4 ⫺ 5x2 ⫹ 7

and

g⬙ 共x兲 ⫽ 12x2 ⫺ 10.

Make a general statement about the relationship of the point of inflection of a function and the second derivative of the function. Because a point of inflection occurs where the concavity of a graph changes, it must be true that at such points the sign of f ⬙ changes. So, to locate possible points of inflection, you only need to determine the values of x for which f ⬙ 共x兲 ⫽ 0 or for which f ⬙ 共x兲 does not exist. This parallels the procedure for locating the relative extrema of f by determining the critical numbers of f. Property of Points of Inflection

If 共c, f 共c兲兲 is a point of inflection of the graph of f, then either f ⬙ 共c兲 ⫽ 0 or f ⬙ 共c兲 is undefined.

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SECTION 3.3 f(x) = x 4 + x 3 − 3x 2 + 1

Example 3

Concavity and the Second-Derivative Test

229

Finding Points of Inflection

y

Discuss the concavity of the graph of f and find its points of inflection. 2

f 共x兲 ⫽ x 4 ⫹ x3 ⫺ 3x2 ⫹ 1

( ) 1 7 , 2 16

−3

−1

1

2

−1

(−1, − 2) −2 −3 −4 −5

FIGURE 3.25 Inflection

Begin by finding the second derivative of f.

SOLUTION

Two Points of

x

f 共x兲 ⫽ f⬘共x兲 ⫽ f ⬙ 共x兲 ⫽ ⫽

x 4 ⫹ x3 ⫺ 3x2 ⫹ 1 4x3 ⫹ 3x2 ⫺ 6x 12x2 ⫹ 6x ⫺ 6 6共2x ⫺ 1兲共x ⫹ 1兲

Write original function. Find first derivative. Find second derivative. Factor.

From this, you can see that the possible points of inflection occur at x ⫽ 12 and x ⫽ ⫺1. After testing the intervals 共⫺ ⬁, ⫺1兲, 共⫺1, 12 兲, and 共12, ⬁兲, you can determine that the graph is concave upward on 共⫺ ⬁, ⫺1兲, concave downward on 共⫺1, 12 兲, and concave upward on 共12, ⬁兲. Because the concavity changes at x ⫽ ⫺1 and x ⫽ 12, you can conclude that the graph has points of inflection at these x-values, as shown in Figure 3.25.

✓CHECKPOINT 3 Discuss the concavity of the graph of f and find its points of inflection. f 共x兲 ⫽ x 4 ⫺ 2x3 ⫹ 1



It is possible for the second derivative to be zero at a point that is not a point of inflection. For example, compare the graphs of f 共x兲 ⫽ x3 and g共x兲 ⫽ x 4, as shown in Figure 3.26. Both second derivatives are zero when x ⫽ 0, but only the graph of f has a point of inflection at x ⫽ 0. This shows that before concluding that a point of inflection exists at a value of x for which f ⬙ 共x兲 ⫽ 0, you must test to be certain that the concavity actually changes at that point. y

f (x)

x3

y

1

−1

g(x)

x4

1

1

−1

f⬙ 共0兲 ⫽ 0, and 共0, 0兲 is a point of inflection.

x

−1

1

x

−1

g⬙ 共0兲 ⫽ 0, but 共0, 0兲 is not a point of inflection.

FIGURE 3.26

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230

CHAPTER 3

Applications of the Derivative

The Second-Derivative Test y

f (c)

The second derivative can be used to perform a simple test for relative minima and relative maxima. If f is a function such that f⬘共c兲 ⫽ 0 and the graph of f is concave upward at x ⫽ c, then f 共c兲 is a relative minimum of f. Similarly, if f is a function such that f⬘共c兲 ⫽ 0 and the graph of f is concave downward at x ⫽ c, then f 共c兲 is a relative maximum of f, as shown in Figure 3.27.

0 Concave downward

x

c Relative maximum

Second-Derivative Test

Let f⬘共c兲 ⫽ 0, and let f ⬙ exist on an open interval containing c. 1. If f ⬙ 共c兲 > 0, then f 共c兲 is a relative minimum.

y

f (c)

2. If f ⬙ 共c兲 < 0, then f 共c兲 is a relative maximum.

0 Concave upward x

c Relative minimum

FIGURE 3.27

3. If f ⬙ 共c兲 ⫽ 0, then the test fails. In such cases, you can use the FirstDerivative Test to determine whether f 共c兲 is a relative minimum, a relative maximum, or neither.

Example 4

Using the Second-Derivative Test

Find the relative extrema of f 共x兲 ⫽ ⫺3x5 ⫹ 5x3. SOLUTION

Begin by finding the first derivative of f.

f⬘共x兲 ⫽ ⫺15x 4 ⫹ 15x 2 ⫽ 15x2共1 ⫺ x2兲

y

Relative maximum (1, 2)

2

From this derivative, you can see that x ⫽ 0, x ⫽ ⫺1, and x ⫽ 1 are the only critical numbers of f. Using the second derivative f ⬙ 共x兲 ⫽ ⫺60x3 ⫹ 30x

1

you can apply the Second-Derivative Test, as shown. x

(0, 0)

−2

2

−1

(−1, − 2) Relative minimum

−2

f (x)

FIGURE 3.28

3x 5

5x 3

Point

Sign of f ⬙ 共x兲

Conclusion

共⫺1, ⫺2兲 共0, 0兲 共1, 2兲

f ⬙ 共⫺1兲 ⫽ 30 > 0 f ⬙ 共0兲 ⫽ 0 f ⬙ 共1兲 ⫽ ⫺30 < 0

Relative minimum Test fails. Relative maximum

Because the test fails at 共0, 0兲, you can apply the First-Derivative Test to conclude that the point 共0, 0兲 is neither a relative minimum nor a relative maximum—a test for concavity would show that this point is a point of inflection. The graph of f is shown in Figure 3.28.

✓CHECKPOINT 4 Find all relative extrema of f 共x) ⫽ x 4 ⫺ 4x3 ⫹ 1.



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SECTION 3.3

Concavity and the Second-Derivative Test

231

Extended Application: Diminishing Returns y

In economics, the notion of concavity is related to the concept of diminishing returns. Consider a function

Concave downward

Input

Output (in dollars)

Output

y ⫽ f 共x兲 Concave upward

Point of diminishing returns

a

c

x

b

where x measures input (in dollars) and y measures output (in dollars). In Figure 3.29, notice that the graph of this function is concave upward on the interval 共a, c兲 and is concave downward on the interval 共c, b兲. On the interval 共a, c兲, each additional dollar of input returns more than the previous input dollar. By contrast, on the interval 共c, b兲, each additional dollar of input returns less than the previous input dollar. The point 共c, f 共c兲兲 is called the point of diminishing returns. An increased investment beyond this point is usually considered a poor use of capital.

Input (in dollars)

FIGURE 3.29

Example 5

By increasing its advertising cost x (in thousands of dollars) for a product, a company discovers that it can increase the sales y (in thousands of dollars) according to the model

Diminishing Returns

Sales (in thousands of dollars)

y

1 3 y = − 10 x + 6x 2 + 400

y⫽⫺

3600 3200 2800

SOLUTION

2000 1600 1200

Concave upward

800 400

x 20

30

0 ≤ x ≤ 40.

Begin by finding the first and second derivatives.

y⬘ ⫽ 12x ⫺

Point of diminishing returns

10

1 3 x ⫹ 6x2 ⫹ 400, 10

Find the point of diminishing returns for this product.

Concave downward

2400

Exploring Diminishing Returns

40

Advertising cost (in thousands of dollars)

FIGURE 3.30

3x2 10

First derivative

3x Second derivative 5 The second derivative is zero only when x ⫽ 20. By testing the intervals 共0, 20兲 and 共20, 40兲, you can conclude that the graph has a point of diminishing returns when x ⫽ 20, as shown in Figure 3.30. So, the point of diminishing returns for this product occurs when $20,000 is spent on advertising. y⬙ ⫽ 12 ⫺

CONCEPT CHECK

✓CHECKPOINT 5 Find the point of diminishing returns for the model below, where R is the revenue (in thousands of dollars) and x is the advertising cost (in thousands of dollars). R⫽

1 共450x2 ⫺ x3兲, 20,000

0 ≤ x ≤ 300



1. Let f be differentiable on an open interval I. If the graph of f is concave upward on I, what can you conclude about the behavior of f⬘ on the interval? 2. Let f be a function whose second derivative exists on an open interval I and f⬙ 冇x冈 > 0 for all x in I. Is f concave upward or concave downward on I ? 3. Let f⬘ 冇c冈 ⴝ 0, and let f⬙ exist on an open interval containing c. According to the Second-Derivative Test, what are the possible classifications for f冇c冈? 4. A newspaper headline states that “The rate of growth of the national deficit is decreasing.” What does this mean? What does it imply about the graph of the deficit as a function of time?

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232

CHAPTER 3

Skills Review 3.3

Applications of the Derivative The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 2.4, 2.6, and 3.1.

In Exercises 1– 6, find the second derivative of the function. 1. f 共x兲 ⫽ 4x 4 ⫺ 9x3 ⫹ 5x ⫺ 1

2. g共s兲 ⫽ 共s2 ⫺ 1兲共s2 ⫺ 3s ⫹ 2兲

3. g共x兲 ⫽ 共x2 ⫹ 1兲 4

4. f 共x兲 ⫽ 共x ⫺ 3兲4兾3

5. h共x兲 ⫽

4x ⫹ 3 5x ⫺ 1

6. f 共x兲 ⫽

2x ⫺ 1 3x ⫹ 2

In Exercises 7–10, find the critical numbers of the function. 7. f 共x兲 ⫽ 5x3 ⫺ 5x ⫹ 11 9. g共t兲 ⫽

16 ⫹ t

8. f 共x兲 ⫽ x 4 ⫺ 4x3 ⫺ 10

t2

10. h共x兲 ⫽

Exercises 3.3

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–8, analytically find the open intervals on which the graph is concave upward and those on which it is concave downward. 1. y ⫽

x2

2. y ⫽

⫺x⫺2

x2 ⫺ 1 3. f 共x兲 ⫽ 2x ⫹ 1 5. f 共x兲 ⫽

⫺x3



3x2

⫺2

6. f 共x兲 ⫽

7. y ⫽ ⫺x3 ⫹ 6x2 ⫺ 9x ⫺ 1

9. f 共x兲 ⫽ 6x ⫺ x2 11. f 共x兲 ⫽



27.

y

y

12. f 共x兲 ⫽ x 4 ⫺ 4x3 ⫹ 2

1

29.

2

14. f 共x兲 ⫽ x ⫹

15. f 共x兲 ⫽ 冪x2 ⫹ 1

16. f 共x兲 ⫽ 冪2x2 ⫹ 6

17. f 共x兲 ⫽ 冪9 ⫺ x2

18. f 共x兲 ⫽ 冪4 ⫺ x 2

19. f 共x兲 ⫽

8 x2 ⫹ 2

20. f 共x兲 ⫽

18 x2 ⫹ 3

21. f 共x兲 ⫽

x x⫺1

22. f 共x兲 ⫽

x x2 ⫺ 1

In Exercises 23–26, use a graphing utility to estimate graphically all relative extrema of the function. 1 1 1 23. f 共x兲 ⫽ 2 x 4 ⫺ 3 x 3 ⫺ 2 x 2

1 1 24. f 共x兲 ⫽ ⫺ 3x 5 ⫺ 2x 4 ⫹ x

25. f 共x兲 ⫽ 5 ⫹ 3x2 ⫺ x3

26. f 共x兲 ⫽ 3x3 ⫹ 5x2 ⫺ 2

f )x)

x

y

1

30. y

2

x

y

f )x) y

4 x

13. f 共x兲 ⫽ x2兾3 ⫺ 3

y

f )x)

8. y ⫽ x5 ⫹ 5x 4 ⫺ 40x2

10. f 共x兲 ⫽ 共x ⫺ 5兲2 ⫹ 7x

28.

y

x2 2 x ⫹1

In Exercises 9–22, find all relative extrema of the function. Use the Second-Derivative Test when applicable. 5x2

In Exercises 27–30, state the signs of f⬘ 冇x冈 and f⬙ 冇x冈 on the interval 冇0, 2冈.

x2 ⫹ 4 4. f 共x兲 ⫽ 4 ⫺ x2

24 x2 ⫹ 12

x3

x 4 ⫺ 50x2 8

1

2

x

f )x)

1

2

x

In Exercises 31–38, find the point(s) of inflection of the graph of the function. 31. f 共x兲 ⫽ x3 ⫺ 9x2 ⫹ 24x ⫺ 18 32. f 共x兲 ⫽ x共6 ⫺ x兲2 33. f 共x兲 ⫽ 共x ⫺ 1兲3共x ⫺ 5兲 34. f 共x兲 ⫽ x 4 ⫺ 18x2 ⫹ 5 35. g共x兲 ⫽ 2x 4 ⫺ 8x3 ⫹ 12x2 ⫹ 12x 36. f 共x兲 ⫽ ⫺4x3 ⫺ 8x2 ⫹ 32

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SECTION 3.3 37. h共x兲 ⫽ 共x ⫺ 2兲3共x ⫺ 1兲 38. f 共t兲 ⫽ 共1 ⫺ t兲共t ⫺ 4兲共t ⫺ 4兲 2

In Exercises 39–50, use a graphing utility to graph the function and identify all relative extrema and points of inflection. 39. f 共x兲 ⫽ x3 ⫺ 12x

40. f 共x兲 ⫽ x 3 ⫺ 3x

41. f 共x兲 ⫽ x3 ⫺ 6x2 ⫹ 12x

3 42. f 共x兲 ⫽ x3 ⫺ 2x2 ⫺ 6x

43. f 共x兲 ⫽

1 4 4x

4

45. g共x兲 ⫽ 共x ⫺ 2兲共x ⫹ 1兲

46. g共x兲 ⫽ 共x ⫺ 6兲共x ⫹ 2兲3

47. g共x兲 ⫽ x冪x ⫹ 3

48. g共x兲 ⫽ x冪9 ⫺ x

2

49. f 共x兲 ⫽

4 1 ⫹ x2

50. f 共x兲 ⫽

2 x2 ⫺ 1

f⬘共x兲 > 0 if x < 3

f⬘共3兲 ⫽ 0

f⬘共3兲 is undefined.

f⬘共x兲 > 0 if x > 3

f⬘共x兲 < 0 if x > 3

f⬘⬘共x兲 > 0

f⬘⬘ 共x兲 > 0, x ⫽ 3

53. f 共0兲 ⫽ f 共2兲 ⫽ 0

f⬘共x兲 < 0 if x < 1

f⬘共1兲 ⫽ 0

f⬘共1兲 ⫽ 0

f⬘共x兲 < 0 if x > 1

f⬘共x兲 > 0 if x > 1

f⬘⬘共x兲 < 0

f⬘⬘共x兲 > 0

y

3 2 3

1

2

−2

1 −1

1

x

−1

1

2

4

x

−2 −3

In Exercises 57– 60, you are given f⬘. Find the intervals on which (a) f⬘ 冇x冈 is increasing or decreasing and (b) the graph of f is concave upward or concave downward. (c) Find the relative extrema and inflection points of f. (d) Then sketch a graph of f. 57. f⬘共x兲 ⫽ 2x ⫹ 5

58. f⬘共x兲 ⫽ 3x2 ⫺ 2

59. f⬘共x兲 ⫽ ⫺x2 ⫹ 2x ⫺ 1

60. f⬘共x兲 ⫽ x2 ⫹ x ⫺ 6

Productivity In Exercises 65 and 66, consider a college student who works from 7 P.M. to 11 P.M. assembling mechanical components. The number N of components assembled after t hours is given by the function. At what time is the student assembling components at the greatest rate? 20t 2 4 ⫹ t2

0 ≤ t ≤ 4

, 0 ≤ t ≤ 4

Sales Growth In Exercises 67 and 68, find the time t in years when the annual sales x of a new product are increasing at the greatest rate. Use a graphing utility to verify your results.

y

56.

Average Cost In Exercises 63 and 64, you are given the total cost of producing x units. Find the production level that minimizes the average cost per unit. Use a graphing utility to verify your results.

66. N ⫽

In Exercises 55 and 56, use the graph to sketch the graph of f⬘. Find the intervals on which (a) f⬘ 冇x冈 is positive, (b) f⬘ 冇x冈 is negative, (c) f⬘ is increasing, and (d) f⬘ is decreasing. For each of these intervals, describe the corresponding behavior of f. 55.

0 ≤ x ≤ 5

65. N ⫽ ⫺0.12t 3 ⫹ 0.54t 2 ⫹ 8.22t,

54. f 共0兲 ⫽ f 共2兲 ⫽ 0

f⬘共x兲 > 0 if x < 1

1 共600x2 ⫺ x3兲, 0 ≤ x ≤ 400 50,000

64. C ⫽ 0.002x3 ⫹ 20x ⫹ 500

52. f 共2兲 ⫽ f 共4兲 ⫽ 0

f⬘共x兲 < 0 if x < 3

61. R ⫽

63. C ⫽ 0.5x2 ⫹ 15x ⫹ 5000

In Exercises 51–54, sketch a graph of a function f having the given characteristics. 51. f 共2兲 ⫽ f 共4兲 ⫽ 0

233

Point of Diminishing Returns In Exercises 61 and 62, identify the point of diminishing returns for the inputoutput function. For each function, R is the revenue and x is the amount spent on advertising. Use a graphing utility to verify your results.

62. R ⫽ ⫺ 49共x3 ⫺ 9x2 ⫺ 27兲,

44. f 共x兲 ⫽ 2x ⫺ 8x ⫹ 3

⫺ 2x

2

Concavity and the Second-Derivative Test

67. x ⫽

10,000t 2 9 ⫹ t2

68. x ⫽

500,000t 2 36 ⫹ t 2

In Exercises 69–72, use a graphing utility to graph f, f⬘, and f⬙ in the same viewing window. Graphically locate the relative extrema and points of inflection of the graph of f. State the relationship between the behavior of f and the signs of f⬘ and f⬙. 69. f 共x兲 ⫽ 12 x3 ⫺ x2 ⫹ 3x ⫺ 5, 关0, 3兴 1 5 1 2 70. f 共x兲 ⫽ ⫺ 20 x ⫺ 12 x ⫺ 13 x ⫹ 1, 关⫺2, 2兴

71. f 共x兲 ⫽

2 , 关⫺3, 3兴 x ⫹1 2

72. f 共x兲 ⫽

x2 , 关⫺3, 3兴 x ⫹1 2

73. Average Cost A manufacturer has determined that the total cost of operating a factory is C C ⫽ 0.5x2 ⫹ 10x ⫹ 7200, where x is the number of units produced. At what level of production will the average cost per unit be minimized? 共The average cost per unit is C兾x.兲 74. Inventory Cost The cost C for ordering and storing x units is C ⫽ 2x ⫹ 300,000兾x. What order size will produce a minimum cost?

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234

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Applications of the Derivative

75. Phishing Phishing is a criminal activity used by an individual or group to fraudulently acquire information by masquerading as a trustworthy person or business in an electronic communication. Criminals create spoof sites on the Internet to trick victims into giving them information. The sites are designed to copy the exact look and feel of a “real” site. A model for the number of reported spoof sites from November 2005 through October 2006 is f 共t兲 ⫽

88.253t3



1116.16t2

⫹ 4541.4t ⫹ 4161, 0 ≤ t ≤ 11

where t represents the number of months since November 2005. (Source: Anti-Phishing Working Group) (a) Use a graphing utility to graph the model on the interval 关0, 11兴. (b) Use the graph in part (a) to estimate the month corresponding to the absolute minimum number of spoof sites. (c) Use the graph in part (a) to estimate the month corresponding to the absolute maximum number of spoof sites.

(d) Sales are steady. (e) Sales are declining, but at a lower rate. (f) Sales have bottomed out and have begun to rise. 78. Medicine N⫽

⫺t 3

The spread of a virus can be modeled by

⫹ 12t 2, 0 ≤ t ≤ 12

where N is the number of people infected (in hundreds), and t is the time (in weeks). (a) What is the maximum number of people projected to be infected? (b) When will the virus be spreading most rapidly? (c) Use a graphing utility to graph the model and to verify your results.

Business Capsule

(d) During approximately which month was the rate of increase of the number of spoof sites the greatest? the least? 76. Dow Jones Industrial Average The graph shows the Dow Jones Industrial Average y on Black Monday, October 19, 1987, where t ⫽ 0 corresponds to 9:30 A.M., when the market opens, and t ⫽ 6.5 corresponds to 4 P.M., the closing time. (Source: Wall Street Journal)

Dow Jones Industrial Average

y 2300 2200 2100 2000 1900 1800 1700

2

3

4

n 1985, Pat Alexander Sanford started Alexander Perry, Inc., in Philadelphia, Pennsylvania. The company specializes in providing interior architecture and space planning to corporations, educational institutions, and private residences. Sanford started the company using about $5000 from her personal savings and a grant from the Women’s Enterprise Center in Philadelphia. The company was incorporated in 1992. Revenues for the company topped $714,000 in 2004 and contracts for 2006 totaled about $6 million. Projected sales are currently expected to approach $10 million.

I

Black Monday

1

Photo courtesy of Pat Alexander Sanford

5

6

7

t

Hours

(a) Estimate the relative extrema and absolute extrema of the graph. Interpret your results in the context of the problem. (b) Estimate the point of inflection of the graph on the interval 关1, 3兴. Interpret your result in the context of the problem. 77. Think About It Let S represent monthly sales of a new digital audio player. Write a statement describing S⬘ and S⬙ for each of the following. (a) The rate of change of sales is increasing. (b) Sales are increasing, but at a greater rate. (c) The rate of change of sales is steady.

79. Research Project Use your school’s library, the Internet, or some other reference source to research the financial history of a small company like the one above. Gather the data on the company’s costs and revenues over a period of time, and use a graphing utility to graph a scatter plot of the data. Fit models to the data. Do the models appear to be concave upward or downward? Do they appear to be increasing or decreasing? Discuss the implications of your answers.

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SECTION 3.4

Optimization Problems

235

Section 3.4 ■ Solve real-life optimization problems.

Optimization Problems

Solving Optimization Problems One of the most common applications of calculus is the determination of optimum (minimum or maximum) values. Before learning a general method for solving optimization problems, consider the next example.

Example 1

h

A manufacturer wants to design an open box that has a square base and a surface area of 108 square inches, as shown in Figure 3.31. What dimensions will produce a box with a maximum volume? SOLUTION

x

x

F I G U R E 3 . 3 1 Open Box with Square Base: S ⫽ x2 ⫹ 4xh ⫽ 108

Finding the Maximum Volume

Because the base of the box is square, the volume is

V ⫽ x 2 h.

Primary equation

This equation is called the primary equation because it gives a formula for the quantity to be optimized. The surface area of the box is S ⫽ 共area of base兲 ⫹ 共area of four sides兲 108 ⫽ x2 ⫹ 4xh.

Secondary equation

Because V is to be optimized, it helps to express V as a function of just one variable. To do this, solve the secondary equation for h in terms of x to obtain h⫽

108 ⫺ x 2 4x

and substitute into the primary equation. V ⫽ x2h ⫽ x2

冢1084x⫺ x 冣 ⫽ 27x ⫺ 41 x 2

3

Function of one variable

Before finding which x-value yields a maximum value of V, you need to determine the feasible domain of the function. That is, what values of x make sense in the problem? Because x must be nonnegative and the area of the base 共A ⫽ x2兲 is at most 108, you can conclude that the feasible domain is 0 ≤ x ≤ 冪108.

Feasible domain

Using the techniques described in the first three sections of this chapter, you can determine that 共on the interval 0 ≤ x ≤ 冪108 兲 this function has an absolute maximum when x ⫽ 6 inches and h ⫽ 3 inches.

✓CHECKPOINT 1 Use a graphing utility to graph the volume function V ⫽ 27x ⫺ 14 x3 on 0 ≤ x ≤ 冪108 from Example 1. Verify that the function has an absolute maximum when x ⫽ 6. What is the maximum volume? ■

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236

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Applications of the Derivative

In studying Example 1, be sure that you understand the basic question that it asks. Some students have trouble with optimization problems because they are too eager to start solving the problem by using a standard formula. For instance, in Example 1, you should realize that there are infinitely many open boxes having 108 square inches of surface area. You might begin to solve this problem by asking yourself which basic shape would seem to yield a maximum volume. Should the box be tall, squat, or nearly cubical? You might even try calculating a few volumes, as shown in Figure 3.32, to see if you can get a good feeling for what the optimum dimensions should be. Volume = 74 14 Volume = 92

STUDY TIP Remember that you are not ready to begin solving an optimization problem until you have clearly identified what the problem is. Once you are sure you understand what is being asked, you are ready to begin considering a method for solving the problem.

1

Volume = 103 34

3

3 × 3 × 84

3

4 × 4 × 54 Volume = 108

6×6×3

FIGURE 3.32

5 × 5 × 4 20 Volume = 88

3

8 × 8 × 18

Which box has the greatest volume?

There are several steps in the solution of Example 1. The first step is to sketch a diagram and identify all known quantities and all quantities to be determined. The second step is to write a primary equation for the quantity to be optimized. Then, a secondary equation is used to rewrite the primary equation as a function of one variable. Finally, calculus is used to determine the optimum value. These steps are summarized below. STUDY TIP When performing Step 5, remember that to determine the maximum or minimum value of a continuous function f on a closed interval, you need to compare the values of f at its critical numbers with the values of f at the endpoints of the interval. The greatest of these values is the desired maximum and the least is the desired minimum.

Guidelines for Solving Optimization Problems

1. Identify all given quantities and all quantities to be determined. If possible, make a sketch. 2. Write a primary equation for the quantity that is to be maximized or minimized. (A summary of several common formulas is given in Appendix D.) 3. Reduce the primary equation to one having a single independent variable. This may involve the use of a secondary equation that relates the independent variables of the primary equation. 4. Determine the feasible domain of the primary equation. That is, determine the values for which the stated problem makes sense. 5. Determine the desired maximum or minimum value by the calculus techniques discussed in Sections 3.1 through 3.3.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 3.4

Example 2 Algebra Review For help on the algebra in Example 2, see Example 1(b) in the Chapter 3 Algebra Review, on page 283.

Optimization Problems

237

Finding a Minimum Sum

The product of two positive numbers is 288. Minimize the sum of the second number and twice the first number. SOLUTION

1. Let x be the first number, y the second, and S the sum to be minimized. 2. Because you want to minimize S, the primary equation is S ⫽ 2x ⫹ y.

Primary equation

3. Because the product of the two numbers is 288, you can write the secondary equation as

TECHNOLOGY After you have written the primary equation as a function of a single variable, you can estimate the optimum value by graphing the function. For instance, the graph of S ⫽ 2x ⫹

S ⫽ 2x ⫹

288 x

Function of one variable

4. Because the numbers are positive, the feasible domain is x > 0.

Feasible domain

5. To find the minimum value of S, begin by finding its critical numbers.

288 x

dS 288 ⫽2⫺ 2 dx x

shown below indicates that the minimum value of S occurs when x is about 12.

0⫽2⫺

Find derivative of S.

288 x2

Set derivative equal to 0.

x2 ⫽ 144 x ⫽ ± 12

120

0

Simplify. Critical numbers

Choosing the positive x-value, you can use the First-Derivative Test to conclude that S is decreasing on the interval 共0, 12兲 and increasing on the interval 共12, ⬁兲, as shown in the table. So, x ⫽ 12 yields a minimum, and the two numbers are

Relative minimum when x ≈ 12

0

Secondary equation xy ⫽ 288 288 y⫽ . x Using this result, you can rewrite the primary equation as a function of one variable.

24

x ⫽ 12

and

y⫽

288 ⫽ 24. 12

Interval

0 < x < 12

12 < x
0 dx

S is decreasing.

S is increasing.

dS dx

Conclusion



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238

CHAPTER 3

Applications of the Derivative

Example 3 y

Finding a Minimum Distance

Find the points on the graph of y ⫽ 4 ⫺ x2

y = 4 − x2

that are closest to 共0, 2兲. 3

SOLUTION

(x, y)

d

1. Figure 3.33 indicates that there are two points at a minimum distance from the point 共0, 2兲.

(0, 2)

2. You are asked to minimize the distance d. So, you can use the Distance Formula to obtain a primary equation.

1

−1

d=

1

x

(x − 0)2 + (y − 2)2

FIGURE 3.33

d ⫽ 冪共x ⫺ 0兲2 ⫹ 共 y ⫺ 2兲2

Primary equation

3. Using the secondary equation y ⫽ 4 ⫺ x2, you can rewrite the primary equation as a function of a single variable. d ⫽ 冪x2 ⫹ 共4 ⫺ x2 ⫺ 2兲2 ⫽ 冪x 4 ⫺ 3x 2 ⫹ 4

Substitute 4 ⫺ x 2 for y. Simplify.

Because d is smallest when the expression under the radical is smallest, you simplify the problem by finding the minimum value of f 共x兲 ⫽ x 4 ⫺ 3x2 ⫹ 4. 4. The domain of f is the entire real line. 5. To find the minimum value of f 共x兲, first find the critical numbers of f. f⬘共x兲 ⫽ 4x3 ⫺ 6x 0 ⫽ 4x3 ⫺ 6x 0 ⫽ 2x 共2x2 ⫺ 3兲

Find derivative of f. Set derivative equal to 0. Factor.

x ⫽ 0, x ⫽ 冪 32, x ⫽ ⫺ 冪 32

✓CHECKPOINT 3 Find the points on the graph of y ⫽ 4 ⫺ x2 that are closest to 共0, 3兲. ■

Algebra Review For help on the algebra in Example 3, see Example 1(c) in the Chapter 3 Algebra Review, on page 283.

Critical numbers

By the First-Derivative Test, you can conclude that x ⫽ 0 yields a relative maximum, whereas both 冪3兾2 and ⫺ 冪3兾2 yield a minimum. So, on the graph of y ⫽ 4 ⫺ x2, the points that are closest to the point 共0, 2兲 are

共冪32 , 52 兲

and

共⫺冪 32, 52 兲.

STUDY TIP To confirm the result in Example 3, try computing the distances between several points on the graph of y ⫽ 4 ⫺ x2 and the point 共0, 2兲. For instance, the distance between 共1, 3兲 and 共0, 2兲 is d ⫽ 冪共0 ⫺ 1兲2 ⫹ 共2 ⫺ 3兲2 ⫽ 冪2 ⬇ 1.414. Note that this is greater than the distance between 共冪3兾2, 5兾2兲 and 共0, 2兲, which is d ⫽ 冪共 0 ⫺

冪32 兲 ⫹ 共2 ⫺ 52 兲 ⫽ 冪74 ⬇ 1.323. 2

2

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SECTION 3.4

Example 4

Optimization Problems

239

Finding a Minimum Area

A rectangular page will contain 24 square inches of print. The margins at the top and bottom of the page are 112 inches wide. The margins on each side are 1 inch wide. What should the dimensions of the page be to minimize the amount of paper used? SOLUTION 1 in.

y

1. A diagram of the page is shown in Figure 3.34.

1 in.

2. Letting A be the area to be minimized, the primary equation is

1

12 in.

y

A ⫽ 共x ⫹ 3兲共 y ⫹ 2兲.

3. The printed area inside the margins is given by 24 ⫽ xy.

Printing

Primary equation

x

Secondary equation

Solving this equation for y produces

x

y⫽

24 . x

By substituting this into the primary equation, you obtain Margin

1 12

in.

A = (x + 3)(y + 2)

FIGURE 3.34

冢24x ⫹ 2冣 24 ⫹ 2x ⫽ 共x ⫹ 3兲冢 冣 x

A ⫽ 共x ⫹ 3兲



2x2 30x 72 ⫹ ⫹ x x x

⫽ 2x ⫹ 30 ⫹

72 . x

Write as a function of one variable. Rewrite second factor as a single fraction. Multiply and separate into terms. Simplify.

4. Because x must be positive, the feasible domain is x > 0. 5. To find the minimum area, begin by finding the critical numbers of A. dA 72 ⫽2⫺ 2 dx x 0⫽2⫺

✓CHECKPOINT 4 A rectangular page will contain 54 square inches of print. The margins at the top and bottom of the page are 112 inches wide. The margins on each side are 1 inch wide. What should the dimensions of the page be to minimize the amount of paper used? ■

72 x2 2 x ⫽ 36 x ⫽ ±6

⫺2 ⫽ ⫺

72 x2

Find derivative of A. Set derivative equal to 0. Subtract 2 from each side. Simplify. Critical numbers

Because x ⫽ ⫺6 is not in the feasible domain, you only need to consider the critical number x ⫽ 6. Using the First-Derivative Test, it follows that A is a minimum when x ⫽ 6. So, the dimensions of the page should be x ⫹ 3 ⫽ 6 ⫹ 3 ⫽ 9 inches by y ⫹ 2 ⫽

24 ⫹ 2 ⫽ 6 inches. 6

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240

CHAPTER 3

Applications of the Derivative

As applications go, the four examples described in this section are fairly simple, and yet the resulting primary equations are quite complicated. Real-life applications often involve equations that are at least as complex as these four. Remember that one of the main goals of this course is to enable you to use the power of calculus to analyze equations that at first glance seem formidable. Also remember that once you have found the primary equation, you can use the graph of the equation to help solve the problem. For instance, the graphs of the primary equations in Examples 1 through 4 are shown in Figure 3.35. V 120

S

3 V = 27x − x 4 (6, 108)

120

100

100

80

80

60

60

40

40

20

20 2

4

6

3

9 12 15 18

x

A

d

6 5 4 3 3 , 2

6

Example 2

x 4 − 3x 2 + 4

(−

288 x

(12, 48)

x

8 10 12

Example 1 d=

S = 2x +

7 4

(

(

1

−3 −2 −1

1

3 , 2

2

3

7 4

x

(

Example 3

80 70 60 50 40 30 20 10

(6, 54) A = 30 + 2x +

72 x

3 6 9 12 15 18 21

x

Example 4

FIGURE 3.35

CONCEPT CHECK 1. Complete the following: In an optimization problem, the formula that represents the quantity to be optimized is called the _____ _____. 2. Explain what is meant by the term feasible domain. 3. Explain the difference between a primary equation and a secondary equation. 4. In your own words, state the guidelines for solving an optimization problem.

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SECTION 3.4

Skills Review 3.4

241

Optimization Problems

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Section 3.1.

In Exercises 1– 4, write a formula for the written statement. 1. The sum of one number and half a second number is 12

2. The product of one number and twice another is 24.

3. The area of a rectangle is 24 square units.

4. The distance between two points is 10 units.

In Exercises 5–10, find the critical numbers of the function. 5. y ⫽ x 2 ⫹ 6x ⫺ 9 8. y ⫽ 3x ⫹

6. y ⫽ 2x3 ⫺ x2 ⫺ 4x

96 x2

9. y ⫽

7. y ⫽ 5x ⫹

x2 ⫹ 1 x

Exercises 3.4

10. y ⫽

125 x

x x2 ⫹ 9

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 6, find two positive numbers satisfying the given requirements. 1. The sum is 120 and the product is a maximum. 2. The sum is S and the product is a maximum. 3. The sum of the first and twice the second is 36 and the product is a maximum. 4. The sum of the first and twice the second is 100 and the product is a maximum.

12. Area A dairy farmer plans to enclose a rectangular pasture adjacent to a river. To provide enough grass for the herd, the pasture must contain 180,000 square meters. No fencing is required along the river. What dimensions will use the least amount of fencing? 13. Maximum Volume (a) Verify that each of the rectangular solids shown in the figure has a surface area of 150 square inches. (b) Find the volume of each solid.

5. The product is 192 and the sum is a minimum.

(c) Determine the dimensions of a rectangular solid (with a square base) of maximum volume if its surface area is 150 square inches.

6. The product is 192 and the sum of the first plus three times the second is a minimum. 3

3

In Exercises 7 and 8, find the length and width of a rectangle that has the given perimeter and a maximum area. 7. Perimeter: 100 meters

8. Perimeter: P units

In Exercises 9 and 10, find the length and width of a rectangle that has the given area and a minimum perimeter. 9. Area: 64 square feet

5

5 6

6

5

3.25

10. Area: A square centimeters

11. Maximum Area A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure). What dimensions should be used so that the enclosed area will be a maximum?

y x

11

14. Maximum Volume Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 337.5 square centimeters. 15. Minimum Cost A storage box with a square base must have a volume of 80 cubic centimeters. The top and bottom cost $0.20 per square centimeter and the sides cost $0.10 per square centimeter. Find the dimensions that will minimize cost.

x

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242

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Applications of the Derivative

16. Maximum Area A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.

23. Maximum Area A rectangle is bounded by the x- and y-axes and the graph of y ⫽ 共6 ⫺ x兲兾2 (see figure). What length and width should the rectangle have so that its area is a maximum? y

y

y=

4

6−x 2

2

1

2

3

4

5

x

x x

(1, 2)

6

(x, 0) 1

Figure for 23

17. Minimum Surface Area A net enclosure for golf practice is open at one end (see figure). The volume of the enclosure is 83 13 cubic meters. Find the dimensions that require the least amount of netting.

(0, y)

1

1

y

x

y

3

(x, y)

2

x 2

4

2

3

4

x

Figure for 24

24. Minimum Length A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point 共1, 2兲 (see figure). (a) Write the length L of the hypotenuse as a function of x. (b) Use a graphing utility to approximate x graphically such that the length of the hypotenuse is a minimum. (c) Find the vertices of the triangle such that its area is a minimum. 25. Maximum Area and the semicircle

A rectangle is bounded by the x-axis

y ⫽ 冪25 ⫺ x2 x Figure for 17

6 − 2x

x

Figure for 18

(see figure). What length and width should the rectangle have so that its area is a maximum? y

18. Volume An open box is to be made from a six-inch by six-inch square piece of material by cutting equal squares from the corners and turning up the sides (see figure). Find the volume of the largest box that can be made. 19. Volume An open box is to be made from a two-foot by three-foot rectangular piece of material by cutting equal squares from the corners and turning up the sides. Find the volume of the largest box that can be made in this manner. 20. Maximum Yield A home gardener estimates that 16 apple trees will have an average yield of 80 apples per tree. But because of the size of the garden, for each additional tree planted the yield will decrease by four apples per tree. How many trees should be planted to maximize the total yield of apples? What is the maximum yield? 21. Area A rectangular page is to contain 36 square inches of print. The margins at the top and bottom and on each side are to be 1 12 inches. Find the dimensions of the page that will minimize the amount of paper used. 22. Area A rectangular page is to contain 30 square inches of print. The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page such that the least amount of paper is used.

6

y=

25 − x 2 (x, y)

−4

−2

2

4

x

26. Area Find the dimensions of the largest rectangle that can be inscribed in a semicircle of radius r. (See Exercise 25.) 27. Volume You are designing a soft drink container that has the shape of a right circular cylinder. The container is supposed to hold 12 fluid ounces (1 fluid ounce is approximately 1.80469 cubic inches). Find the dimensions that will use a minimum amount of construction material. 28. Minimum Cost An energy drink container of the shape described in Exercise 27 must have a volume of 16 fluid ounces. The cost per square inch of constructing the top and bottom is twice the cost of constructing the sides. Find the dimensions that will minimize cost.

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SECTION 3.4 In Exercises 29–32, find the points on the graph of the function that are closest to the given point. 29. f 共x兲 ⫽ x2,

冢2, 12冣

30. f 共x兲 ⫽ 共x ⫹ 1兲2, 共5, 3兲 31. f 共x兲 ⫽ 冪x, 共4, 0兲 32. f 共x兲 ⫽ 冪x ⫺ 8, 共2, 0兲 33. Maximum Volume A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package with maximum volume. Assume that the package’s dimensions are x by x by y (see figure).

Optimization Problems

243

39. Maximum Area An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200-meter running track. Find the dimensions that will make the area of the rectangular region as large as possible. 40. Farming A strawberry farmer will receive $30 per bushel of strawberries during the first week of harvesting. Each week after that, the value will drop $0.80 per bushel. The farmer estimates that there are approximately 120 bushels of strawberries in the fields, and that the crop is increasing at a rate of four bushels per week. When should the farmer harvest the strawberries to maximize their value? How many bushels of strawberries will yield the maximum value? What is the maximum value of the strawberries? 41. Beam Strength A wooden beam has a rectangular cross section of height h and width w (see figure). The strength S of the beam is directly proportional to its width and the square of its height. What are the dimensions of the strongest beam that can be cut from a round log of diameter 24 inches? (Hint: S ⫽ kh 2 w, where k > 0 is the proportionality constant.)

x x

y

w

34. Minimum Surface Area A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic inches. Find the radius of the cylinder that produces the minimum surface area. 35. Minimum Cost An industrial tank of the shape described in Exercise 34 must have a volume of 3000 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimensions that will minimize cost.

24

h

42. Area Four feet of wire is to be used to form a square and a circle.

36. Minimum Area The sum of the perimeters of a circle and a square is 16. Find the dimensions of the circle and square that produce a minimum total area.

(a) Express the sum of the areas of the square and the circle as a function A of the side of the square x.

37. Minimum Area The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and square that produce a minimum total area.

(c) Use a graphing utility to graph A on its domain.

38. Minimum Time You are in a boat 2 miles from the nearest point on the coast. You are to go to point Q, located 3 miles down the coast and 1 mile inland (see figure). You can row at a rate of 2 miles per hour and you can walk at a rate of 4 miles per hour. Toward what point on the coast should you row in order to reach point Q in the least time?

(b) What is the domain of A? (d) How much wire should be used for the square and how much for the circle in order to enclose the least total area? the greatest total area? 43. Profit The profit P 共in thousands of dollars兲 for a company spending an amount s 共in thousands of dollars兲 on advertising is P⫽⫺

1 3 s ⫹ 6s2 ⫹ 400. 10

(a) Find the amount of money the company should spend on advertising in order to yield a maximum profit. 2 mi x

(b) Find the point of diminishing returns.

3−x 1 mi 3 mi

Q

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244

CHAPTER 3

Applications of the Derivative

Mid-Chapter Quiz

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Take this quiz as you would take a quiz in class. When you are done, check your work against the answers given in the back of the book. In Exercises 1–3, find the critical numbers of the function and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. 1. f 共x兲 ⫽ x2 ⫺ 6x ⫹ 1

2. f 共x兲 ⫽ 2x 3 ⫹ 12x 2

3. f 共x兲 ⫽

1 x2 ⫹ 2

In Exercises 4 – 6, use a graphing utility to graph the function. Then find all relative extrema of the function. 4. f 共x兲 ⫽ x 3 ⫹ 3x 2 ⫺ 5

5. f 共x兲 ⫽ x 4 ⫺ 8x 2 ⫹ 3

6. f 共x兲 ⫽ 2x2兾3

In Exercises 7–9, find the absolute extrema of the function on the closed interval. 7. f 共x兲 ⫽ x 2 ⫹ 2x ⫺ 8, 关⫺2, 1兴 9. f 共x兲 ⫽

8. f 共x兲 ⫽ x 3 ⫺ 27x, 关⫺4, 4兴

x , 关0, 2兴 x2 ⫹ 1

In Exercises 10 and 11, find the point(s) of inflection of the graph of the function. Then determine the open intervals on which the graph of the function is concave upward or concave downward. 10. f 共x兲 ⫽ x 3 ⫺ 6x 2 ⫹ 7x

11. f 共x兲 ⫽ x 4 ⫺ 24x 2

In Exercises 12 and 13, Use the Second-Derivative Test to find all relative extrema of the function. x2 ⫹ 1 12. f 共x兲 ⫽ 2x 3 ⫹ 3x 2 ⫺ 12x ⫹ 16 13. f 共x兲 ⫽ x 14. By increasing its advertising cost x for a product, a company discovers that it can increase the sales S according to the model S⫽ y

y x

Figure for 15

1 共360x 2 ⫺ x 3兲, 3600

0 ≤ x ≤ 240

where x and S are in thousands of dollars. Find the point of diminishing returns for this product. 15. A gardener has 200 feet of fencing to enclose a rectangular garden adjacent to a river (see figure). No fencing is needed along the river. What dimensions should be used so that the area of the garden will be a maximum? 16. The resident population P (in thousands) of the District of Columbia from 1999 through 2005 can be modeled by P ⫽ 0.2694t3 ⫺ 2.048t2 ⫺ 0.73t ⫹ 571.9 where ⫺1 ≤ t ≤ 5 and t ⫽ 0 corresponds to 2000.

(Source: U.S. Census Bureau)

(a) During which year, from 1999 through 2005, was the population the greatest? the least? (b) During which year(s) was the population increasing? decreasing?

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SECTION 3.5

Business and Economics Applications

245

Section 3.5 ■ Solve business and economics optimization problems.

Business and Economics Applications

■ Find the price elasticity of demand for demand functions. ■ Recognize basic business terms and formulas.

Optimization in Business and Economics The problems in this section are primarily optimization problems. So, the fivestep procedure used in Section 3.4 is an appropriate strategy to follow.

Example 1

Finding the Maximum Revenue

A company has determined that its total revenue (in dollars) for a product can be modeled by R ⫽ ⫺x3 ⫹ 450x2 ⫹ 52,500x where x is the number of units produced (and sold). What production level will yield a maximum revenue?

Maximum Revenue R

R = −x 3 + 450x 2 + 52,500x

Revenue (in dollars)

35,000,000

SOLUTION

(350, 30,625,000)

30,000,000

1. A sketch of the revenue function is shown in Figure 3.36.

25,000,000

2. The primary equation is the given revenue function.

20,000,000

R ⫽ ⫺x3 ⫹ 450x2 ⫹ 52,500x

15,000,000

Primary equation

3. Because R is already given as a function of one variable, you do not need a secondary equation.

10,000,000 5,000,000 200

400

600

x

Number of units

F I G U R E 3 . 3 6 Maximum revenue occurs when dR兾dx ⫽ 0.

4. The feasible domain of the primary equation is 0 ≤ x ≤ 546.

Feasible domain

This is determined by finding the x-intercepts of the revenue function, as shown in Figure 3.36. 5. To maximize the revenue, find the critical numbers. dR ⫽ ⫺3x2 ⫹ 900x ⫹ 52,500 ⫽ 0 dx ⫺3共x ⫺ 350兲共x ⫹ 50兲 ⫽ 0 x ⫽ 350, x ⫽ ⫺50

Set derivative equal to 0. Factor. Critical numbers

The only critical number in the feasible domain is x ⫽ 350. From the graph of the function, you can see that the production level of 350 units corresponds to a maximum revenue.

✓CHECKPOINT 1 Find the number of units that must be produced to maximize the revenue function R ⫽ ⫺x3 ⫹ 150x2 ⫹ 9375x. What is the maximum revenue? ■

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246

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To study the effects of production levels on cost, economists use the average cost function C, which is defined as C⫽

C x

Average cost function

where C ⫽ f 共x兲 is the total cost function and x is the number of units produced.

Example 2

Finding the Minimum Average Cost

A company estimates that the cost (in dollars) of producing x units of a product can be modeled by C ⫽ 800 ⫹ 0.04x ⫹ 0.0002x2. Find the production level that minimizes the average cost per unit. SOLUTION

1. C represents the total cost, x represents the number of units produced, and C represents the average cost per unit.

STUDY TIP To see that x ⫽ 2000 corresponds to a minimum average cost in Example 2, try evaluating C for several values of x. For instance, when x ⫽ 400, the average cost per unit is C ⫽ $2.12, but when x ⫽ 2000, the average cost per unit is C ⫽ $0.84.

2. The primary equation is C⫽

C . x

Primary equation

3. Substituting the given equation for C produces C⫽ ⫽

800 ⫹ 0.04x ⫹ 0.0002x2 x

Substitute for C.

800 ⫹ 0.04 ⫹ 0.0002x. x

Function of one variable

4. The feasible domain for this function is x > 0. Minimum Average Cost

Average cost (in dollars)

C 2.00

C=

800 + 0.04 + 0.0002x x $

Feasible domain

5. You can find the critical numbers as shown. dC 800 ⫽ ⫺ 2 ⫹ 0.0002 ⫽ 0 dx x

1.50

0.0002 ⫽

1.00 0.50

1000 2000 3000 4000

x

Number of units

F I G U R E 3 . 3 7 Minimum average cost occurs when d C 兾dx ⫽ 0.

800 x2

800 0.0002 x2 ⫽ 4,000,000 x ⫽ ± 2000 x2 ⫽

Set derivative equal to 0.

Multiply each side by x2 and divide each side by 0.0002.

Critical numbers

By choosing the positive value of x and sketching the graph of C, as shown in Figure 3.37, you can see that a production level of x ⫽ 2000 minimizes the average cost per unit.

✓CHECKPOINT 2 Find the production level that minimizes the average cost per unit for the cost function C ⫽ 400 ⫹ 0.05x ⫹ 0.0025x2. ■

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SECTION 3.5

Example 3

Business and Economics Applications

247

Finding the Maximum Revenue

A business sells 2000 units of a product per month at a price of $10 each. It can sell 250 more items per month for each $0.25 reduction in price. What price per unit will maximize the monthly revenue? SOLUTION

1. Let x represent the number of units sold in a month, let p represent the price per unit, and let R represent the monthly revenue. 2. Because the revenue is to be maximized, the primary equation is R ⫽ xp.

Primary equation

3. A price of p ⫽ $10 corresponds to x ⫽ 2000, and a price of p ⫽ $9.75 corresponds to x ⫽ 2250. Using this information, you can use the point-slope form to create the demand equation. Maximum Revenue

10 ⫺ 9.75 共x ⫺ 2000兲 2000 ⫺ 2250 p ⫺ 10 ⫽ ⫺0.001共x ⫺ 2000兲

p ⫺ 10 ⫽

R

Revenue (in dollars)

40,000

(6000, 36,000)

30,000

p ⫽ ⫺0.001x ⫹ 12

20,000 10,000

Point-slope form Simplify. Secondary equation

Substituting this value into the revenue equation produces R = 12x − 0.001x2 4000

8000

12,000

x

Number of units

FIGURE 3.38

STUDY TIP In Example 3, the revenue function was written as a function of x. It could also have been written as a function of p. That is, R ⫽ 1000共12p ⫺ p2兲. By finding the critical numbers of this function, you can determine that the maximum revenue occurs when p ⫽ 6.

R ⫽ x共⫺0.001x ⫹ 12兲 ⫽ ⫺0.001x2 ⫹ 12x.

Substitute for p. Function of one variable

4. The feasible domain of the revenue function is 0 ≤ x ≤ 12,000.

Feasible domain

5. To maximize the revenue, find the critical numbers. dR ⫽ 12 ⫺ 0.002x ⫽ 0 dx ⫺0.002x ⫽ ⫺12 x ⫽ 6000

Set derivative equal to 0.

Critical number

From the graph of R in Figure 3.38, you can see that this production level yields a maximum revenue. The price that corresponds to this production level is p ⫽ 12 ⫺ 0.001x ⫽ 12 ⫺ 0.001共6000兲 ⫽ $6.

Demand function Substitute 6000 for x. Price per unit

✓CHECKPOINT 3 Find the price per unit that will maximize the monthly revenue for the business in Example 3 if it can sell only 200 more items per month for each $0.25 reduction in price. ■

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248

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Example 4

Algebra Review For help on the algebra in Example 4, see Example 2(b) in the Chapter 3 Algebra Review, on page 284.

Finding the Maximum Profit

The marketing department of a business has determined that the demand for a product can be modeled by p⫽

50 .

冪x

The cost of producing x units is given by C ⫽ 0.5x ⫹ 500. What price will yield a maximum profit? SOLUTION

Maximum Profit P 900

Profit (in dollars)

800

P = 50

1. Let R represent the revenue, P the profit, p the price per unit, x the number of units, and C the total cost of producing x units.

x − 0.5x − 500

2. Because you are maximizing the profit, the primary equation is

700 600 500

P ⫽ R ⫺ C.

(2500, 750)

3. Because the revenue is R ⫽ xp, you can write the profit function as

400 300 200 100 2000

4000

6000

8000

x

Number of units

FIGURE 3.39

冢 冣

Substitute for R and C. Substitute for p. Function of one variable

5. To maximize the profit, find the critical numbers.

Find the price that will maximize profit for the demand and cost functions. 40 p⫽ and C ⫽ 2x ⫹ 50 冪x

dP 25 ⫺ 0.5 ⫽ 0 ⫽ dx 冪x 冪x ⫽ 50 x ⫽ 2500



Marginal Revenue and Marginal Cost Revenue and cost (in dollars)

P⫽R⫺C ⫽ xp ⫺ 共0.5x ⫹ 500兲 50 ⫽x ⫺ 0.5x ⫺ 500 冪x ⫽ 50冪x ⫺ 0.5x ⫺ 500.

4. The feasible domain of the function is 127 < x ≤ 7872. (When x is less than 127 or greater than 7872, the profit is negative.)

✓CHECKPOINT 4

R = 50 x

Set derivative equal to 0. Isolate x-term on one side. Critical number

From the graph of the profit function shown in Figure 3.39, you can see that a maximum profit occurs when x ⫽ 2500. The price that corresponds to x ⫽ 2500 is p⫽

3500 3000

Primary equation

50 50 50 ⫽ ⫽ ⫽ $1.00. 50 冪x 冪2500

Price per unit

2500 Maximum profit: dR = dC dx dx

2000 1500 1000 500

STUDY TIP To find the maximum profit in Example 4, the equation P ⫽ R ⫺ C was differentiated and set equal to zero. From the equation

C = 0.5x + 500 1000 2000 3000 4000 5000

Number of units

FIGURE 3.40

x

dP dR dC ⫽ ⫺ ⫽0 dx dx dx it follows that the maximum profit occurs when the marginal revenue is equal to the marginal cost, as shown in Figure 3.40.

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SECTION 3.5

One way economists measure the responsiveness of consumers to a change in the price of a product is with price elasticity of demand. For example, a drop in the price of vegetables might result in a much greater demand for vegetables; such a demand is called elastic. On the other hand, the demand for items such as milk and water is relatively unresponsive to changes in price; the demand for such items is called inelastic. More formally, the elasticity of demand is the percent change of a quantity demanded x, divided by the percent change in its price p. You can develop a formula for price elasticity of demand using the approximation

(Source: James Kearl, Principles of Economics)

Absolute Value of Elasticity

Cottonseed oil

6.92

Tomatoes

4.60

Restaurant meals

1.63

Automobiles

1.35

Cable TV

1.20

Beer

1.13

Housing

1.00

Movies

0.87

Clothing

0.60

Cigarettes

0.51

Coffee

0.25

Gasoline

0.15

Newspapers

0.10

Mail

0.05

249

Price Elasticity of Demand

STUDY TIP The list below shows some estimates of elasticities of demand for common products.

Item

Business and Economics Applications

⌬ p dp ⬇ ⌬x dx which is based on the definition of the derivative. Using this approximation, you can write Price elasticity of demand ⫽

rate of change in demand rate of change in price



⌬x兾x ⌬p兾p



p兾x ⌬p兾⌬x



p兾x . dp兾dx

Definition of Price Elasticity of Demand

If p ⫽ f 共x兲 is a differentiable function, then the price elasticity of demand is given by

Which of these items are elastic? Which are inelastic?

␩⫽

p兾x dp兾dx

where ␩ is the lowercase Greek letter eta. For a given price, the demand is elastic if ␩ > 1, the demand is inelastic if ␩ < 1, and the demand has unit elasticity if ␩ ⫽ 1.

ⱍⱍ

R

Elastic dR >0 dx

ⱍⱍ

Inelastic dR 1, x

0 < x < 64

Elastic

which implies that the demand is elastic when 0 < x < 64. For x-values in the interval 共64, 144兲,

ⱍ␩ⱍ ⫽

(b)

Multiply numerator and denominator by ⫺

is x ⫽ 64. So, the demand is of unit elasticity when x ⫽ 64. For x-values in the interval 共0, 64兲,

250

Number of units

Substitute for p兾x and dp兾dx.

ⱍⱍ

400

25 50 75 100 125 150

Formula for price elasticity of demand

The demand is of unit elasticity when ␩ ⫽ 1. In the interval 关0, 144兴, the only solution of the equation

Revenue Function of a Product

300

p兾x dp兾dx 18 ⫺ 1.5冪x x ⫽ ⫺3 4冪x ⫺24冪x ⫹ 2x ⫽ x 24冪x ⫽⫺ ⫹ 2. x

␩⫽

21 18 15 12 9 6 3



24冪x ⫹ 2 < 1, 64 < x < 144 x

Inelastic

which implies that the demand is inelastic when 64 < x < 144. b. From part (a), you can conclude that the revenue function R is increasing on the open interval 共0, 64兲, is decreasing on the open interval 共64, 144兲, and is a maximum when x ⫽ 64, as indicated in Figure 3.42(b).

STUDY TIP In the discussion of price elasticity of demand, the price is assumed to decrease as the quantity demanded increases. So, the demand function p ⫽ f 共x兲 is decreasing and dp兾dx is negative.

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SECTION 3.5

Business and Economics Applications

251

Business Terms and Formulas This section concludes with a summary of the basic business terms and formulas used in this section. A summary of the graphs of the demand, revenue, cost, and profit functions is shown in Figure 3.43. Summary of Business Terms and Formulas x ⫽ number of units produced (or sold) p ⫽ price per unit R ⫽ total revenue from selling x units ⫽ xp C ⫽ total cost of producing x units P ⫽ total profit from selling x units ⫽ R ⫺ C C C ⫽ average cost per unit ⫽ x p

R

␩ ⫽ price elasticity of demand ⫽ 共 p兾x兲兾共dp兾dx兲 dR兾dx ⫽ marginal revenue dC兾dx ⫽ marginal cost dP兾dx ⫽ marginal profit

Elastic demand

Inelastic demand

p = f (x)

x

x

Demand function

Revenue function

Quantity demanded increases as price decreases.

The low prices required to sell more units eventually result in a decreasing revenue. P

C

Fixed cost

Maximum profit Break-even point x

x

Negative of fixed cost

Cost function

Profit function

The total cost to produce x units includes the fixed cost.

The break-even point occurs when R ⫽ C.

FIGURE 3.43

CONCEPT CHECK C 1. In the average cost function C ⴝ , what does C represent? What does x x represent? 2. After a drop in the price of tomatoes, the demand for tomatoes increased. This is an example of what type of demand? 3. Even though the price of gasoline rose, the demand for gasoline was the same. This is an example of what type of demand? 4. Explain how price elasticity of demand is related to the total revenue function.

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252

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Skills Review 3.5

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 0.2, 0.3, 0.5, and 2.3.

In Exercises 1– 4, evaluate the expression for x ⴝ 150.

ⱍ ⱍ

1. ⫺



300 ⫹3 x

共20x⫺1兾2兲兾x 3. ⫺10x⫺3兾2

ⱍ ⱍ

2. ⫺





600 ⫹2 5x

共4000兾x2兲兾x 4. ⫺8000x⫺3



In Exercises 5–10, find the marginal revenue, marginal cost, or marginal profit. 5. C ⫽ 650 ⫹ 1.2x ⫹ 0.003x2 7. R ⫽ 14x ⫺

6. P ⫽ 0.01x2 ⫹ 11x

x 2000 2

8. R ⫽ 3.4x ⫺

9. P ⫽ ⫺0.7x2 ⫹ 7x ⫺ 50

10. C ⫽ 1700 ⫹ 4.2x ⫹ 0.001x3

Exercises 3.5

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 4, find the number of units x that produces a maximum revenue R. 1. R ⫽ 800x ⫺ 0.2x2

2. R ⫽ 48x2 ⫺ 0.02x3

3. R ⫽ 400x ⫺ x2

4. R ⫽ 30x2兾3 ⫺ 2x

In Exercises 5 – 8, find the number of units x that produces the minimum average cost per unit C. 5. C ⫽

0.125x2

⫹ 20x ⫹ 5000

7. C ⫽ 2x2 ⫹ 255x ⫹ 5000 8. C ⫽ 0.02x3 ⫹ 55x2 ⫹ 1380 In Exercises 9 –12, find the price per unit p that produces the maximum profit P. Cost Function

13. C ⫽ 2x2 ⫹ 5x ⫹ 18

14. C ⫽ x3 ⫺ 6x2 ⫹ 13x

15. Maximum Profit A commodity has a demand function modeled by p ⫽ 100 ⫺ 0.5x, and a total cost function modeled by C ⫽ 40x ⫹ 37.5. (a) What price yields a maximum profit? (b) When the profit is maximized, what is the average cost per unit? 16. Maximum Profit How would the answer to Exercise 15 change if the marginal cost rose from $40 per unit to $50 per unit? In other words, rework Exercise 15 using the cost function C ⫽ 50x ⫹ 37.5.

6. C ⫽ 0.001x3 ⫹ 5x ⫹ 250

9. C ⫽ 100 ⫹ 30x

x2 1500

Demand Function p ⫽ 90 ⫺ x

10. C ⫽ 0.5x ⫹ 500

50 p⫽ 冪x

11. C ⫽ 8000 ⫹ 50x ⫹ 0.03x2

p ⫽ 70 ⫺ 0.01x

12. C ⫽ 35x ⫹ 500

p ⫽ 50 ⫺ 0.1冪x

Average Cost In Exercises 13 and 14, use the cost function to find the production level for which the average cost is a minimum. For this production level, show that the marginal cost and average cost are equal. Use a graphing utility to graph the average cost function and verify your results.

Maximum Profit In Exercises 17 and 18, find the amount s of advertising that maximizes the profit P. (s and P are measured in thousands of dollars.) Find the point of diminishing returns. 17. P ⫽ ⫺2s3 ⫹ 35s2 ⫺ 100s ⫹ 200 18. P ⫽ ⫺0.1s3 ⫹ 6s2 ⫹ 400 19. Maximum Profit The cost per unit of producing a type of digital audio player is $60. The manufacturer charges $90 per unit for orders of 100 or less. To encourage large orders, however, the manufacturer reduces the charge by $0.10 per player for each order in excess of 100 units. For instance, an order of 101 players would be $89.90 per player, an order of 102 players would be $89.80 per player, and so on. Find the largest order the manufacturer should allow to obtain a maximum profit.

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SECTION 3.5

Business and Economics Applications

253

20. Maximum Profit A real estate office handles a 50-unit apartment complex. When the rent is $580 per month, all units are occupied. For each $40 increase in rent, however, an average of one unit becomes vacant. Each occupied unit requires an average of $45 per month for service and repairs. What rent should be charged to obtain a maximum profit?

Elasticity In Exercises 27–32, find the price elasticity of demand for the demand function at the indicated x-value. Is the demand elastic, inelastic, or of unit elasticity at the indicated x-value? Use a graphing utility to graph the revenue function, and identify the intervals of elasticity and inelasticity.

21. Maximum Revenue When a wholesaler sold a product at $40 per unit, sales were 300 units per week. After a price increase of $5, however, the average number of units sold dropped to 275 per week. Assuming that the demand function is linear, what price per unit will yield a maximum total revenue?

27. p ⫽ 600 ⫺ 5x

x ⫽ 30

28. p ⫽ 400 ⫺ 3x

x ⫽ 20

29. p ⫽ 5 ⫺ 0.03x

x ⫽ 100

30. p ⫽ 20 ⫺ 0.0002x

x ⫽ 30

500 31. p ⫽ x⫹2

x ⫽ 23

22. Maximum Profit Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on the money. Furthermore, the bank can reinvest the money at 12% simple interest. Find the interest rate the bank should pay to maximize its profit. 23. Minimum Cost A power station is on one side of a river that is 0.5 mile wide, and a factory is 6 miles downstream on the other side of the river (see figure). It costs $18 per foot to run overland power lines and $25 per foot to run underwater power lines. Write a cost function for running the power lines from the power station to the factory. Use a graphing utility to graph your function. Estimate the value of x that minimizes the cost. Explain your results.

Demand Function

32. p ⫽

Quantity Demanded

100 ⫹2 x2

x ⫽ 10

33. Elasticity The demand function for a product is given by p ⫽ 20 ⫺ 0.02x,

0 < x < 1000.

(a) Find the price elasticity of demand when x ⫽ 560. (b) Find the values of x and p that maximize the total revenue. (c) For the value of x found in part (b), show that the price elasticity of demand has unit elasticity. 34. Elasticity The demand function for a product is given by p ⫽ 800 ⫺ 4x,

x 6−x

Factory

(a) Find the price elasticity of demand when x ⫽ 150.

1 2

(b) Find the values of x and p that maximize the total revenue.

Power station

(c) For the value of x found in part (b), show that the price elasticity of demand has unit elasticity. 35. Minimum Cost The shipping and handling cost C of a manufactured product is modeled by

River

24. Minimum Cost An offshore oil well is 1 mile off the coast. The oil refinery is 2 miles down the coast. Laying pipe in the ocean is twice as expensive as laying it on land. Find the most economical path for the pipe from the well to the oil refinery. Minimum Cost In Exercises 25 and 26, find the speed v, in miles per hour, that will minimize costs on a 110-mile delivery trip. The cost per hour for fuel is C dollars, and the driver is paid W dollars per hour. (Assume there are no costs other than wages and fuel.) 25. Fuel cost: C ⫽

v2 300

Driver: W ⫽ $12

0 < x < 200.

26. Fuel cost: C ⫽

v2 500

Driver: W ⫽ $9.50

C⫽4

冢25x ⫺ x ⫺x 10冣, 2

0 < x < 10

where C is measured in thousands of dollars and x is the number of units shipped (in hundreds). Find the shipment size that minimizes the cost. (Hint: Use the root feature of a graphing utility.) 36. Minimum Cost The ordering and transportation cost C of the components used in manufacturing a product is modeled by C⫽8

x ⫺ , 冢2500 x x ⫺ 100 冣 2

0 < x < 100

where C is measured in thousands of dollars and x is the order size in hundreds. Find the order size that minimizes the cost. (Hint: Use the root feature of a graphing utility.)

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254

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Applications of the Derivative

37. MAKE A DECISION: REVENUE The demand for a car wash is x ⫽ 600 ⫺ 50p, where the current price is $5. Can revenue be increased by lowering the price and thus attracting more customers? Use price elasticity of demand to determine your answer. 38. Revenue Repeat Exercise 37 for a demand function of x ⫽ 800 ⫺ 40p. 39. Sales The sales S (in billions of dollars per year) for Procter & Gamble for the years 2001 through 2006 can be modeled by S ⫽ 1.09312t2 ⫺ 1.8682t ⫹ 39.831,

1 ≤ t ≤ 6

where t represents the year, with t ⫽ 1 corresponding to 2001. (Source: Procter & Gamble Company) (a) During which year, from 2001 through 2006, were Procter & Gamble’s sales increasing most rapidly? (b) During which year were the sales increasing at the lowest rate?

42. Demand A demand function is modeled by x ⫽ a兾pm, where a is a constant and m > 1. Show that ␩ ⫽ ⫺m. In other words, show that a 1% increase in price results in an m% decrease in the quantity demanded. 43. Think About It Throughout this text, it is assumed that demand functions are decreasing. Can you think of a product that has an increasing demand function? That is, can you think of a product that becomes more in demand as its price increases? Explain your reasoning, and sketch a graph of the function. 44. Extended Application To work an extended application analyzing the sales per share for Lowe’s from 1990 through 2005, visit this text’s website at college.hmco.com. (Data Source: Lowe’s Companies)

Business Capsule

(c) Find the rate of increase or decrease for each year in parts (a) and (b). (d) Use a graphing utility to graph the sales function. Then use the zoom and trace features to confirm the results in parts (a), (b), and (c). 40. Revenue The revenue R (in millions of dollars per year) for Papa John’s from 1996 to 2005 can be modeled by R⫽

⫺485.0 ⫹ 116.68t , 6 ≤ t ≤ 15 1 ⫺ 0.12t ⫹ 0.0097t 2

Photo courtesy of Jim Bell

where t represents the year, with t ⫽ 6 corresponding to 1996. (Source: Papa John’s Int’l.) (a) During which year, from 1996 through 2005, was Papa John’s revenue the greatest? the least? (b) During which year was the revenue increasing at the greatest rate? decreasing at the greatest rate? (c) Use a graphing utility to graph the revenue function, and confirm your results in parts (a) and (b). 41. Match each graph with the function it best represents— a demand function, a revenue function, a cost function, or a profit function. Explain your reasoning. (The graphs are labeled a – d.) y 35,000

20,000

b

15,000 10,000 5,000

c d 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000

I

45. Research Project Choose an innovative product like the one described above. Use your school’s library, the Internet, or some other reference source to research the history of the product or service. Collect data about the revenue that the product or service has generated, and find a mathematical model of the data. Summarize your findings.

a

30,000 25,000

llinois native Jim Bell moved to California in 1996 to pursue his dream of working in the skateboarding industry. After a string of sales jobs with several skate companies, Bell started San Diego-based Jim Bell Skateboard Ramps in 2004 with an initial cash outlay of $50. His custom-built skateboard ramp business brought in sales of $250,000 the following year. His latest product, the U-Built-It Skateboard Ramp, is expected to nearly double his annual sales. Bell marketed his new product by featuring it at trade shows. He backed it up by showing pictures of the hundreds of ramps he has built. So, Bell was able to prove the demand existed, as well as the quality and customer satisfaction his work boasted.

x

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SECTION 3.6

Asymptotes

255

Section 3.6 ■ Find the vertical asymptotes of functions and find infinite limits.

Asymptotes

■ Find the horizontal asymptotes of functions and find limits at infinity. ■ Use asymptotes to answer questions about real-life situations.

Vertical Asymptotes and Infinite Limits y

In the first three sections of this chapter, you studied ways in which you can use calculus to help analyze the graph of a function. In this section, you will study another valuable aid to curve sketching: the determination of vertical and horizontal asymptotes. Recall from Section 1.5, Example 10, that the function

8 6

3 f (x) = x−2

4

−∞

2



2

2

−2

3 x−2 as x

3 x−2 as x

−4 −6

4

6

8

x = 2 is a vertical asymptote.

−8

FIGURE 3.44

f 共x兲 ⫽

x

3 x⫺2

is unbounded as x approaches 2 (see Figure 3.44). This type of behavior is described by saying that the line x ⫽ 2 is a vertical asymptote of the graph of f. The type of limit in which f 共x兲 approaches infinity (or negative infinity) as x approaches c from the left or from the right is an infinite limit. The infinite limits for the function f 共x兲 ⫽ 3兾共x ⫺ 2兲 can be written as lim

3 ⫽ ⫺⬁ x⫺2

lim

3 ⫽ . x⫺2 ⬁

x→2⫺

and x→2⫹

Definition of Vertical Asymptote

If f 共x兲 approaches infinity (or negative infinity) as x approaches c from the right or from the left, then the line x ⫽ c is a vertical asymptote of the graph of f.

TECHNOLOGY When you use a graphing utility to graph a function that has a vertical asymptote, the utility may try to connect separate branches of the graph. For instance, the figure at the right shows the graph of 3 f 共x兲 ⫽ x⫺2

This line is not part of the graph of the function.

5

9

−6

The graph of the function has two branches. −5

on a graphing calculator.

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256

CHAPTER 3

Applications of the Derivative

TECHNOLOGY Use a spreadsheet or table to verify the results shown in Example 1. (Consult the user’s manual of a spreadsheet software program for specific instructions on how to create a table.) For instance, in Example 1(a), notice that the values of f 共x兲 ⫽ 1兾共x ⫺ 1兲 decrease and increase without bound as x gets closer and closer to 1 from the left and the right. x Approaches 1 from the Left

f 共x兲 ⫽ 1兾共x ⫺ 1兲

x 0

⫺1

0.9

⫺10

0.99

⫺100

0.999

⫺1000

0.9999

⫺10,000

One of the most common instances of a vertical asymptote is the graph of a rational function—that is, a function of the form f 共x兲 ⫽ p共x兲兾q共x兲, where p共x兲 and q共x兲 are polynomials. If c is a real number such that q共c兲 ⫽ 0 and p共c兲 ⫽ 0, the graph of f has a vertical asymptote at x ⫽ c. Example 1 shows four cases.

Example 1 Find each limit.

Limit from the left 1 ⫽ ⫺⬁ x⫺1

b. lim⫺

⫺1 ⫽ x⫺1

c. lim⫺

⫺1 ⫽ ⫺⬁ 共x ⫺ 1兲2

x→1⫹

d. lim⫺

1 ⫽ 共x ⫺ 1兲2 ⬁

x→1⫹

x→1

x→1

x→1

x→1

1 ⫽ x⫺1 ⬁

See Figure 3.45(a).



lim⫹

⫺1 ⫽ ⫺⬁ x⫺1

See Figure 3.45(b).

lim

⫺1 ⫽ ⫺⬁ 共x ⫺ 1兲2

See Figure 3.45(c).

lim

1 ⫽ 共x ⫺ 1兲2 ⬁

See Figure 3.45(d).

x→1

y

2

2

1

1

2

1

1.1

10

1.01

100

1.001

1000

−2

1.0001

10,000

−3

2 −1

lim

x

1

3

x

−1

−2

−2

f (x) =

−3

lim

x

1

x

2 −1

1 x−1

f (x) =

1 = −∞ x−1

1 = ∞ x−1

(a)

lim

x

1

−1 = ∞ x−1

lim

x

1

−1 x−1

−1 = −∞ x−1

(b) y

y

Find each limit. a. Limit from the left 1 lim x→2 ⫺ x ⫺ 2 Limit from the right 1 lim x→2 ⫹ x ⫺ 2 b. Limit from the left ⫺1 lim x→⫺3 ⫺ x ⫹ 3 Limit from the right ⫺1 lim x→⫺3 ⫹ x ⫹ 3 ■

lim

x→1⫹

y

f 共x兲 ⫽ 1兾共x ⫺ 1兲

✓CHECKPOINT 1

Limit from the right

a. lim⫺

x Approaches 1 from the Right

x

Finding Infinite Limits

2

2

f (x) =

1

−2

−1 (x − 1)2

2

1 x

−2

−1

2

−1

−1

−2

−2

−3

−3

lim

x

1

−1 = −∞ (x − 1)2

(c)

f (x) =

lim

x

1

3

x

1 (x − 1)2

1 =∞ (x − 1)2

(d)

FIGURE 3.45

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SECTION 3.6

Asymptotes

257

Each of the graphs in Example 1 has only one vertical asymptote. As shown in the next example, the graph of a rational function can have more than one vertical asymptote. y

f (x) =

x+2 x 2 − 2x

Example 2

Finding Vertical Asymptotes

Find the vertical asymptotes of the graph of f 共x兲 ⫽ 1 −2

−1

1

3

4

x

5

−1

x⫹2 . x 2 ⫺ 2x

SOLUTION The possible vertical asymptotes correspond to the x-values for which

the denominator is zero.

x2 ⫺ 2x ⫽ 0 x共x ⫺ 2兲 ⫽ 0 x ⫽ 0, x ⫽ 2

−2 −3 −4

Set denominator equal to 0. Factor. Zeros of denominator

Because the numerator of f is not zero at either of these x-values, you can conclude that the graph of f has two vertical asymptotes—one at x ⫽ 0 and one at x ⫽ 2, as shown in Figure 3.46.

F I G U R E 3 . 4 6 Vertical Asymptotes at x ⫽ 0 and x ⫽ 2

✓CHECKPOINT 2 Find the vertical asymptote(s) of the graph of y

f 共x兲 ⫽

x⫹4 . x2 ⫺ 4x



4

2

Undefined when x = 2

Example 3

Finding Vertical Asymptotes

Find the vertical asymptotes of the graph of −6

−4

2

x

f 共x兲 ⫽

−2

SOLUTION

Vertical Asymptote

✓CHECKPOINT 3 Find the vertical asymptotes of the graph of x2 ⫹ 4x ⫹ 3 . x2 ⫺ 9

x 2 ⫹ 2x ⫺ 8 x2 ⫺ 4 共x ⫹ 4兲共x ⫺ 2兲 ⫽ 共x ⫹ 2兲共x ⫺ 2兲 共x ⫹ 4兲共x ⫺ 2兲 ⫽ 共x ⫹ 2兲共x ⫺ 2兲

f 共x兲 ⫽

2 f (x) = x +2 2x − 8 x −4

f 共x兲 ⫽

First factor the numerator and denominator. Then divide out like

factors.

−4

FIGURE 3.47 at x ⫽ ⫺2

x 2 ⫹ 2x ⫺ 8 . x2 ⫺ 4





x⫹4 , x⫹2

x⫽2

Write original function. Factor numerator and denominator. Divide out like factors. Simplify.

For all values of x other than x ⫽ 2, the graph of this simplified function is the same as the graph of f. So, you can conclude that the graph of f has only one vertical asymptote. This occurs at x ⫽ ⫺2, as shown in Figure 3.47.

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258

CHAPTER 3

Applications of the Derivative

From Example 3, you know that the graph of f 共x兲 ⫽

x 2 ⫹ 2x ⫺ 8 x2 ⫺ 4

has a vertical asymptote at x ⫽ ⫺2. This implies that the limit of f 共x兲 as x → ⫺2 from the right (or from the left) is either ⬁ or ⫺ ⬁. But without looking at the graph, how can you determine that the limit from the left is negative infinity and the limit from the right is positive infinity? That is, why is the limit from the left lim ⫺

x→⫺2

x 2 ⫹ 2x ⫺ 8 ⫽ ⫺⬁ x2 ⫺ 4

Limit from the left

and why is the limit from the right lim ⫹

x→⫺2

From the left, f )x) approaches positive infinity.

x 2 ⫹ 2x ⫺ 8 ⫽ x2 ⫺ 4

Limit from the right

It is cumbersome to determine these limits analytically, and you may find the graphical method shown in Example 4 to be more efficient.

4

Example 4 4

−4

⬁?

lim⫺

x→1

x 2 ⫺ 3x x⫺1

SOLUTION

FIGURE 3.48

STUDY TIP In Example 4, try evaluating f 共x兲 at x-values that are just barely to the left of 1. You will find that you can make the values of f 共x兲 arbitrarily large by choosing x sufficiently close to 1. For instance, f 共0.99999兲 ⫽ 199,999.

`

Find the limits.

−4

From the right, f )x) approaches negative infinity.

Determining Infinite Limits

f 共x兲 ⫽

and

lim⫹

x→1

x 2 ⫺ 3x x⫺1

Begin by considering the function x 2 ⫺ 3x . x⫺1

Because the denominator is zero when x ⫽ 1 and the numerator is not zero when x ⫽ 1, it follows that the graph of the function has a vertical asymptote at x ⫽ 1. This implies that each of the given limits is either ⬁ or ⫺ ⬁. To determine which, use a graphing utility to graph the function, as shown in Figure 3.48. From the graph, you can see that the limit from the left is positive infinity and the limit from the right is negative infinity. That is, lim⫺

x 2 ⫺ 3x ⫽ x⫺1

lim⫹

x 2 ⫺ 3x ⫽ ⫺ ⬁. x⫺1

x→1



Limit from the left

and x→1

Limit from the right

✓CHECKPOINT 4 Find the limits. lim⫺

x→2

x2 ⫺ 4x x⫺2

and

lim⫹

x→2

x2 ⫺ 4x x⫺2

Then verify your solution by graphing the function.



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SECTION 3.6

259

Asymptotes

Horizontal Asymptotes and Limits at Infinity Another type of limit, called a limit at infinity, specifies a finite value approached by a function as x increases (or decreases) without bound. y

Definition of Horizontal Asymptote

y = L1

If f is a function and L1 and L2 are real numbers, the statements y = f(x)

lim f 共x兲 ⫽ L1 and

x

lim f 共x兲 ⫽ L 2

x→ ⬁

x→ ⫺⬁

denote limits at infinity. The lines y ⫽ L 1 and y ⫽ L 2 are horizontal asymptotes of the graph of f.

y = L2

y

Figure 3.49 shows two ways in which the graph of a function can approach one or more horizontal asymptotes. Note that it is possible for the graph of a function to cross its horizontal asymptote. Limits at infinity share many of the properties of limits discussed in Section 1.5. When finding horizontal asymptotes, you can use the property that

y = f(x) y=L x

lim

x→ ⬁

1 ⫽ 0, xr

r > 0

and

lim

x→ ⫺⬁

1 ⫽ 0, xr

r > 0.

共The second limit assumes that x r is defined when x < 0.兲

FIGURE 3.49

Example 5

Finding Limits at Infinity



Find the limit: lim 5 ⫺ x→ ⬁



2 . x2

SOLUTION

x→ ⬁

10

y = 5 − 22 x

8 6

−4

−2

FIGURE 3.50



2 2 ⫽ lim 5 ⫺ lim 2 x→ ⬁ x→ ⬁ x x2



⫽ lim 5 ⫺ 2 lim x→ ⬁

y = 5 is a horizontal asymptote.

x→ ⬁

lim 关 f 共x兲 ⫺ g共x兲兴 ⫽ lim f 共x兲 ⫺ lim g共x兲

x→ ⬁

1 x2



x→ ⬁

x→ ⬁

lim c f 共x兲 ⫽ c lim f 共x兲

x→ ⬁

x→ ⬁

⫽ 5 ⫺ 2共0兲 ⫽5

4

−6



lim 5 ⫺

y

You can verify this limit by sketching the graph of

2

4

6

x

f 共x兲 ⫽ 5 ⫺

2 x2

as shown in Figure 3.50. Note that the graph has y ⫽ 5 as a horizontal asymptote to the right. By evaluating the limit of f 共x兲 as x → ⫺ ⬁, you can show that this line is also a horizontal asymptote to the left.

✓CHECKPOINT 5



Find the limit: lim 2 ⫹ x→ ⬁



5 . x2



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260

CHAPTER 3

Applications of the Derivative

There is an easy way to determine whether the graph of a rational function has a horizontal asymptote. This shortcut is based on a comparison of the degrees of the numerator and denominator of the rational function. TECHNOLOGY

Horizontal Asymptotes of Rational Functions

Some functions have two horizontal asymptotes: one to the right and one to the left. For instance, try sketching the graph of f 共x兲 ⫽

x 冪x 2

⫹1

Let f 共x兲 ⫽ p共x兲兾q共x兲 be a rational function. 1. If the degree of the numerator is less than the degree of the denominator, then y ⫽ 0 is a horizontal asymptote of the graph of f (to the left and to the right). 2. If the degree of the numerator is equal to the degree of the denominator, then y ⫽ a兾b is a horizontal asymptote of the graph of f (to the left and to the right), where a and b are the leading coefficients of p共x兲 and q共x兲, respectively.

.

What horizontal asymptotes does the function appear to have?

3. If the degree of the numerator is greater than the degree of the denominator, then the graph of f has no horizontal asymptote.

✓CHECKPOINT 6

Example 6

Find the horizontal asymptote of the graph of each function.

Finding Horizontal Asymptotes

Find the horizontal asymptote of the graph of each function. ⫺2x ⫹ 3 3x2 ⫹ 1

a. y ⫽

2x ⫹ 1 4x2 ⫹ 5

a. y ⫽

b. y ⫽

2x2 ⫹ 1 4x2 ⫹ 5

SOLUTION

c. y ⫽

2x3 ⫹ 1 4x2 ⫹ 5

b. y ⫽

⫺2x 2 ⫹ 3 3x 2 ⫹ 1

c. y ⫽

⫺2x 3 ⫹ 3 3x 2 ⫹ 1

a. Because the degree of the numerator is less than the degree of the denominator, y ⫽ 0 is a horizontal asymptote. [See Figure 3.51(a).] b. Because the degree of the numerator is equal to the degree of the denominator, the line y ⫽ ⫺ 23 is a horizontal asymptote. [See Figure 3.51(b).]



c. Because the degree of the numerator is greater than the degree of the denominator, the graph has no horizontal asymptote. [See Figure 3.51(c).] y

y

y

3

3 2 y = − 2x2 + 3 3x + 1

+3 y = − 2x 3x 2 + 1

2 1 −3 −2

−1

−1

3 y = − 2x2 + 3 3x + 1

1

1 1

2

3

−2

(a) y ⫽ 0 is a horizontal asymptote.

x

−1

−1

1

x

−2

(b) y ⫽ ⫺ 23 is a horizontal asymptote.

−3 −2

−1

−1

1

2

3

x

−2

(c) No horizontal asymptote

FIGURE 3.51

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SECTION 3.6

Asymptotes

261

Applications of Asymptotes There are many examples of asymptotic behavior in real life. For instance, Example 7 describes the asymptotic behavior of an average cost function.

Example 7 STUDY TIP In Example 7, suppose that the small business had made an initial investment of $50,000. How would this change the answers to the questions? Would it change the average cost of producing x units? Would it change the limiting average cost per unit?

Modeling Average Cost

A small business invests $5000 in a new product. In addition to this initial investment, the product will cost $0.50 per unit to produce. Find the average cost per unit if 1000 units are produced, if 10,000 units are produced, and if 100,000 units are produced. What is the limit of the average cost as the number of units produced increases? SOLUTION

dollars) by

From the given information, you can model the total cost C (in

C ⫽ 0.5x ⫹ 5000

Total cost function

where x is the number of units produced. This implies that the average cost function is C⫽

C 5000 . ⫽ 0.5 ⫹ x x

Average cost function

If only 1000 units are produced, then the average cost per unit is

Average cost per unit (in dollars)

C ⫽ 0.5 ⫹ C 5.00 4.50 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50

5000 ⫽ $5.50. 1000

Average cost for 1000 units

If 10,000 units are produced, then the average cost per unit is

Average Cost

C ⫽ 0.5 ⫹

5000 ⫽ $1.00. 10,000

Average cost for 10,000 units

If 100,000 units are produced, then the average cost per unit is C=

20,000

C 5000 = 0.5 + x x

60,000

Number of units

F I G U R E 3 . 5 2 As x → ⬁, the average cost per unit approaches $0.50.

C ⫽ 0.5 ⫹ x

5000 ⫽ $0.55. 100,000

Average cost for 100,000 units

As x approaches infinity, the limiting average cost per unit is



lim 0.5 ⫹

x→ ⬁



5000 ⫽ $0.50. x

As shown in Figure 3.52, this example points out one of the major problems of small businesses. That is, it is difficult to have competitively low prices when the production level is low.

✓CHECKPOINT 7 A small business invests $25,000 in a new product. In addition, the product will cost $0.75 per unit to produce. Find the cost function and the average cost function. What is the limit of the average cost function as production increases? ■

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262

CHAPTER 3

Applications of the Derivative

Example 8

Modeling Smokestack Emission

A manufacturing plant has determined that the cost C (in dollars) of removing p% of the smokestack pollutants of its main smokestack is modeled by C⫽

80,000p , 100 ⫺ p

0 ≤ p < 100.

What is the vertical asymptote of this function? What does the vertical asymptote mean to the plant owners? The graph of the cost function is shown in Figure 3.53. From the graph, you can see that p ⫽ 100 is the vertical asymptote. This means that as the plant attempts to remove higher and higher percents of the pollutants, the cost increases dramatically. For instance, the cost of removing 85% of the pollutants is

© Joel W. Rogers/Corbis

SOLUTION

Since the 1980s, industries in the United States have spent billions of dollars to reduce air pollution.

C⫽

80,000共85兲 ⬇ $453,333 100 ⫺ 85

Cost for 85% removal

but the cost of removing 90% is C⫽

80,000共90兲 ⫽ $720,000. 100 ⫺ 90

Cost for 90% removal

Smokestack Emission C 1,000,000 900,000

Cost (in dollars)

800,000

✓CHECKPOINT 8 According to the cost function in Example 8, is it possible to remove 100% of the smokestack pollutants? Why or why not? ■

(90, 720,000)

700,000 600,000 500,000

(85, 453,333)

400,000

80,000p C= 100 − p

300,000 200,000 100,000

p 10

20

30

40

50

60

70

80

90

100

Percent of pollutants removed

FIGURE 3.53

CONCEPT CHECK 1. Complete the following: If f 冇x冈 → ±ⴥ as x → c from the right or the left, then the line x ⴝ c is a _____ _____ of the graph of f. 2. Describe in your own words what is meant by lim f 冇x冈 ⴝ 4. x→ⴥ

3. Describe in your own words what is meant by lim f 冇x冈 ⴝ 2. x→ⴚⴥ

4. Complete the following: Given a rational function f, if the degree of the numerator is less than the degree of the denominator, then _______ is a horizontal asymptote of the graph of f (to the left and to the right).

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 3.6

Skills Review 3.6

263

Asymptotes

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 1.5, 2.3, and 3.5.

In Exercises 1–8, find the limit. 1. lim 共x ⫹ 1兲

2. lim 共3x ⫹ 4兲

2x2 ⫹ x ⫺ 15 3. lim x→⫺3 x⫹3

4. lim

x→2

5. lim ⫹ x→2

x→⫺1

x→2

x 2 ⫺ 5x ⫹ 6 x2 ⫺ 4

3x2 ⫺ 8x ⫹ 4 x⫺2

6. lim⫺ x→1

x 2 ⫺ 6x ⫹ 5 x2 ⫺ 1

8. lim⫹ 共x ⫹ 冪x ⫺ 1 兲

7. lim⫹ 冪x x→0

x→1

In Exercises 9–12, find the average cost and the marginal cost. 9. C ⫽ 150 ⫹ 3x 10. C ⫽ 1900 ⫹ 1.7x ⫹ 0.002x 2 11. C ⫽ 0.005x 2 ⫹ 0.5x ⫹ 1375 12. C ⫽ 760 ⫹ 0.05x

Exercises 3.6

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1– 8, find the vertical and horizontal asymptotes. Write the asymptotes as equations of lines. x2 ⫹ 1 1. f 共x兲 ⫽ x2

5. f 共x兲 ⫽

3 2 1 −1

−3 − 2 − 1 −1

1

2

3

x

3

1

4

5

x

−2

4 3 2

−4 −3 −2 −1 −2 −3 −4

7. f 共x兲 ⫽

−3

1 2 3 4

x

x2 ⫺ 1 2x 2 ⫺ 8

2⫹x 4. f 共x兲 ⫽ 1⫺x

y

y

8. f 共x兲 ⫽

3 2

−2

1

3

x

− 2 −1

−3

3

3

2

2 1

2 3 4

x

x

−1

x2 ⫹ 1 x3 ⫺ 8

y

1

3 2

x

−3 −2 −1 −2 −3 −4

y

x2 ⫺ 2 3. f 共x兲 ⫽ 2 x ⫺x⫺2

⫺4x x2 ⫹ 4 y

4 3 2 1

y

2

6. f 共x兲 ⫽

y

4 2. f 共x兲 ⫽ 共x ⫺ 2兲3

y

3x2 2共x2 ⫹ 1兲

1

3

−1

−2

−2

−3

−3

3

4

5

−2 −3 −4

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

x

264

CHAPTER 3

Applications of the Derivative

In Exercises 9–12, match the function with its graph. Use horizontal asymptotes as an aid. [The graphs are labeled (a)–(d).] y

(a)

y

(b) 2

3

x

1 1 −2

−1

1

2

x

x

2

⫺10 2

2

2x

−3 −2 −1

1 −2 − 1

9. f 共x兲 ⫽

1

2

3

3x 2 ⫹2

11. f 共x兲 ⫽ 2 ⫹

x2 4 x ⫹1

1

2

3

x

(a) h共x兲 ⫽ x x2 ⫹ 2

12. f 共x兲 ⫽ 5 ⫺

x→ ⬁

lim ⫺

x→⫺2

1 共x ⫹ 2兲2

14.

1 x2 ⫹ 1

x→⫺2

2⫹x 16. lim⫹ x→1 1 ⫺ x

x2 17. lim⫺ 2 x→4 x ⫺ 16

x2 18. lim 2 x→4 x ⫹ 16



19. lim⫺ 1 ⫹ x→0

1 x





20. lim⫺ x 2 ⫺ x→0

10 0

101

10 2

10 3

10 4

f 共x兲 x4

f 共x兲 x

(b) h共x兲 ⫽

f 共x兲 x2

(c) h共x兲 ⫽

f 共x兲 x3

30. (a) lim

3 ⫺ 2x 3x 3 ⫺ 1

(b) lim

x2 ⫹ 2 x2 ⫺ 1

(b) lim

3 ⫺ 2x 3x ⫺ 1

(c) lim

x2 ⫹ 2 x⫺1

(c) lim

3 ⫺ 2x2 3x ⫺ 1

x→ ⬁

x→ ⬁

x→ ⬁

x→ ⬁

In Exercises 31– 40, find the limit. 5x3 ⫹ 1 x→ ⬁ 10x ⫺ 3x2 ⫹ 7

4x ⫺ 3 x→ ⬁ 2x ⫹ 1

32. lim

3x x→ ⬁ 4x ⫺ 1

34.

31. lim 1 x

33. lim



In Exercises 21–24, use a graphing utility or spreadsheet software program to complete the table. Then use the result to estimate the limit of f 冇x冈 as x approaches infinity. x

(c) h共x兲 ⫽

x2 ⫹ 2 x3 ⫺ 1

x→ ⬁

1 x⫹2

x⫺4 15. lim⫹ x→3 x ⫺ 3

f 共x兲 x3

29. (a) lim

x→ ⬁

lim ⫺

(b) h共x兲 ⫽

In Exercises 29 and 30, find each limit, if possible.

In Exercises 13–20, find the limit. 13.

10 6

28. f 共x兲 ⫽ 3x2 ⫹ 7

x

10. f 共x兲 ⫽

x2

10 4

26. f 共x兲 ⫽ x ⫺ 冪x共x ⫺ 1兲

冪x2 ⫹ 4

f 共x兲 (a) h共x兲 ⫽ x2

1

2

10 2

f 共x兲

27. f 共x兲 ⫽ 5x 3 ⫺ 3 3

10 0

In Exercises 27 and 28, find lim h冇x冈, if possible.

y

(d)

⫺10 4

⫺10 6

25. f 共x兲 ⫽

−2

y

(c)

1

−1

1

In Exercises 25 and 26, use a graphing utility or a spreadsheet software program to complete the table and use the result to estimate the limit of f 冇x冈 as x approaches infinity and as x approaches negative infinity.

10 5

10 6

f 共x兲

35.

2

lim

x→⫺⬁

5x 2 x⫹3

x→ ⬁

39.

lim

x→⫺⬁



x 3 ⫺ 2x 2 ⫹ 3x ⫹ 1 x 2 ⫺ 3x ⫹ 2

38. lim 共2 ⫺ x⫺3兲

x→ ⬁

2x 3x ⫹ x⫺1 x⫹1

2x2 ⫺ 5x ⫺ 12 x→⫺⬁ 1 ⫺ 6x ⫺ 8x2 lim

36. lim

37. lim 共2x ⫺ x⫺2兲

3

x→ ⬁



40. lim

x→ ⬁

2

冢x 2x⫺ 1 ⫹ x 3x⫹ 1冣

In Exercises 41–58, sketch the graph of the equation. Use intercepts, extrema, and asymptotes as sketching aids. 3x 1⫺x

x⫹1 21. f 共x兲 ⫽ x冪x

2x2 22. f 共x兲 ⫽ x⫹1

41. y ⫽

x2 ⫺ 1 23. f 共x兲 ⫽ 0.02x 2

3x2 24. f 共x兲 ⫽ 0.1x 2 ⫹ 1

43. f 共x兲 ⫽ 45. g共x兲 ⫽

42. y ⫽

x⫺3 x⫺2

x2 ⫹9

44. f 共x兲 ⫽

x x2 ⫹ 4

x2 x 2 ⫺ 16

46. g共x兲 ⫽

x x2 ⫺ 4

x2

47. xy 2 ⫽ 4

48. x 2 y ⫽ 4

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SECTION 3.6 49. y ⫽

2x 1⫺x

51. y ⫽ 1 ⫺ 3x⫺2 53. f 共x兲 ⫽ 55. g共x兲 ⫽ 57. y ⫽

1 x2 ⫺ x ⫺ 2 x2

⫺x⫺2 x⫺2

2x2 ⫺ 6 共x ⫺ 1兲2

50. y ⫽

2x 1 ⫺ x2

52. y ⫽ 1 ⫹ x⫺1 54. f 共x兲 ⫽ 56. g共x兲 ⫽ 58. y ⫽

x⫺2 x 2 ⫺ 4x ⫹ 3 x2

⫺9 x⫹3

x 共x ⫹ 1兲2

59. Cost The cost C (in dollars) of producing x units of a product is C ⫽ 1.35x ⫹ 4570. (a) Find the average cost function C. (b) Find C when x ⫽ 100 and when x ⫽ 1000. (c) What is the limit of C as x approaches infinity? 60. Average Cost A business has a cost (in dollars) of C ⫽ 0.5x ⫹ 500 for producing x units. (a) Find the average cost function C. (b) Find C when x ⫽ 250 and when x ⫽ 1250. (c) What is the limit of C as x approaches infinity? 61. Average Cost The cost function for a certain model of personal digital assistant (PDA) is given by C ⫽ 13.50x ⫹ 45,750, where C is measured in dollars and x is the number of PDAs produced. (a) Find the average cost function C. (b) Find C when x ⫽ 100 and x ⫽ 1000. (c) Determine the limit of the average cost function as x approaches infinity. Interpret the limit in the context of the problem. 62. Average Cost The cost function for a company to recycle x tons of material is given by C ⫽ 1.25x ⫹ 10,500, where C is measured in dollars. (a) Find the average cost function C. (b) Find the average costs of recycling 100 tons of material and 1000 tons of material. (c) Determine the limit of the average cost function as x approaches infinity. Interpret the limit in the context of the problem. 63. Seizing Drugs The cost C (in millions of dollars) for the federal government to seize p% of a type of illegal drug as it enters the country is modeled by C ⫽ 528p兾共100 ⫺ p兲,

0 ≤ p < 100.

(a) Find the costs of seizing 25%, 50%, and 75%. p → 100 ⫺ .

(b) Find the limit of C as Interpret the limit in the context of the problem. Use a graphing utility to verify your result.

265

Asymptotes

64. Removing Pollutants The cost C (in dollars) of removing p% of the air pollutants in the stack emission of a utility company that burns coal is modeled by C ⫽ 80,000p兾共100 ⫺ p兲,

0 ≤ p < 100.

(a) Find the costs of removing 15%, 50%, and 90%. (b) Find the limit of C as p → 100 ⫺ . Interpret the limit in the context of the problem. Use a graphing utility to verify your result. 65. Learning Curve Psychologists have developed mathematical models to predict performance P (the percent of correct responses) as a function of n, the number of times a task is performed. One such model is P⫽

0.5 ⫹ 0.9共n ⫺ 1兲 , 1 ⫹ 0.9共n ⫺ 1兲

0 < n.

(a) Use a spreadsheet software program to complete the table for the model. n

1

2

3

4

5

6

7

8

9

10

P (b) Find the limit as n approaches infinity. (c) Use a graphing utility to graph this learning curve, and interpret the graph in the context of the problem. 66. Biology: Wildlife Management The state game commission introduces 30 elk into a new state park. The population N of the herd is modeled by N ⫽ 关10共3 ⫹ 4t兲兴兾共1 ⫹ 0.1t兲 where t is the time in years. (a) Find the size of the herd after 5, 10, and 25 years. (b) According to this model, what is the limiting size of the herd as time progresses? 67. Average Profit The cost and revenue functions for a product are C ⫽ 34.5x ⫹ 15,000 and R ⫽ 69.9x. (a) Find the average profit function P ⫽ 共R ⫺ C兲兾x. (b) Find the average profits when x is 1000, 10,000, and 100,000. (c) What is the limit of the average profit function as x approaches infinity? Explain your reasoning. 68. Average Profit The cost and revenue functions for a product are C ⫽ 25.5x ⫹ 1000 and R ⫽ 75.5x. (a) Find the average profit function P ⫽

R⫺C . x

(b) Find the average profits when x is 100, 500, and 1000. (c) What is the limit of the average profit function as x approaches infinity? Explain your reasoning.

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266

CHAPTER 3

Applications of the Derivative

Section 3.7

Curve Sketching: A Summary

■ Analyze the graphs of functions. ■ Recognize the graphs of simple polynomial functions.

Summary of Curve-Sketching Techniques

40

−2

5 − 10 200 30

− 10

− 1200

FIGURE 3.54

It would be difficult to overstate the importance of using graphs in mathematics. Descartes’s introduction of analytic geometry contributed significantly to the rapid advances in calculus that began during the mid-seventeenth century. So far, you have studied several concepts that are useful in analyzing the graph of a function. • x-intercepts and y-intercepts (Section 1.2) • Domain and range (Section 1.4) • Continuity (Section 1.6) • Differentiability (Section 2.1) • Relative extrema (Section 3.2) • Concavity (Section 3.3) • Points of inflection (Section 3.3) • Vertical asymptotes (Section 3.6) • Horizontal asymptotes (Section 3.6) When you are sketching the graph of a function, either by hand or with a graphing utility, remember that you cannot normally show the entire graph. The decision as to which part of the graph to show is crucial. For instance, which of the viewing windows in Figure 3.54 better represents the graph of f 共x兲 ⫽ x3 ⫺ 25x2 ⫹ 74x ⫺ 20?

TECHNOLOGY Which of the viewing windows best represents the graph of the function x 3 ⫹ 8x 2 ⫺ 33x f 共x兲 ⫽ ? 5 a. Xmin ⫽ ⫺15, Xmax ⫽ 1, Ymin ⫽ ⫺10, Ymax ⫽ 60 b. Xmin ⫽ ⫺10, Xmax ⫽ 10, Ymin ⫽ ⫺10, Ymax ⫽ 10 c. Xmin ⫽ ⫺13, Xmax ⫽ 5, Ymin ⫽ ⫺10, Ymax ⫽ 60

The lower viewing window gives a more complete view of the graph, but the context of the problem might indicate that the upper view is better. Here are some guidelines for analyzing the graph of a function. Guidelines for Analyzing the Graph of a Function

1. Determine the domain and range of the function. If the function models a real-life situation, consider the context. 2. Determine the intercepts and asymptotes of the graph. 3. Locate the x-values where f⬘共x兲 and f ⬙ 共x兲 are zero or undefined. Use the results to determine where the relative extrema and the points of inflection occur. In these guidelines, note the importance of algebra (as well as calculus) for solving the equations f 共x兲 ⫽ 0, f⬘共x兲 ⫽ 0, and f⬘⬘共x兲 ⫽ 0.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 3.7

Example 1

Curve Sketching: A Summary

267

Analyzing a Graph

Analyze the graph of f 共x兲 ⫽ x3 ⫹ 3x2 ⫺ 9x ⫹ 5. SOLUTION

y

Relative maximum (−3, 32)

(−1, 16) Point of inflection

−6

f (x) = x +

f⬘共x兲 ⫽ 3x2 ⫹ 6x ⫺ 9 ⫽ 3共x ⫺ 1兲共x ⫹ 3兲.

3x 2

FIGURE 3.55

− 9x + 5

First derivative Factored form

So, the critical numbers of f are x ⫽ 1 and x ⫽ ⫺3. The second derivative of f is

(0, 5)

− 10

Factored form

So, the x-intercepts occur when x ⫽ 1 and x ⫽ ⫺5. The derivative is

20

−4 −3 −2 −1

3

Begin by finding the intercepts of the graph. This function factors as

f 共x兲 ⫽ 共x ⫺ 1兲2共x ⫹ 5兲. 30

(−5, 0)

Original function

(1, 0) 2 Relative minimum

x

f ⬙ 共x兲 ⫽ 6x ⫹ 6 ⫽ 6共x ⫹ 1兲

Second derivative Factored form

which implies that the second derivative is zero when x ⫽ ⫺1. By testing the values of f⬘共x兲 and f ⬙ 共x兲, as shown in the table, you can see that f has one relative minimum, one relative maximum, and one point of inflection. The graph of f is shown in Figure 3.55. f 共x兲 x in 共⫺ ⬁, ⫺3兲 x ⫽ ⫺3

32

x in 共⫺3, ⫺1兲 x ⫽ ⫺1

16

x in 共⫺1, 1兲 x⫽1

0

x in 共1, ⬁兲

f⬘ 共x兲

f ⬙ 共x兲





Increasing, concave downward

0



Relative maximum





Decreasing, concave downward



0

Point of inflection





Decreasing, concave upward

0



Relative minimum





Increasing, concave upward

Characteristics of graph

✓CHECKPOINT 1 Analyze the graph of f 共x兲 ⫽ ⫺x 3 ⫹ 3x 2 ⫹ 9x ⫺ 27.



TECHNOLOGY In Example 1, you are able to find the zeros of f, f⬘, and f ⬙ algebraically (by factoring). When this is not feasible, you can use a graphing utility to find the zeros. For instance, the function g共x兲 ⫽ x3 ⫹ 3x2 ⫺ 9x ⫹ 6 is similar to the function in the example, but it does not factor with integer coefficients. Using a graphing utility, you can determine that the function has only one x-intercept, x ⬇ ⫺5.0275.

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268

CHAPTER 3

Applications of the Derivative

Example 2

Analyzing a Graph

Analyze the graph of f 共x兲 ⫽ x4 ⫺ 12x3 ⫹ 48x2 ⫺ 64x. SOLUTION y

f (x) = x 4 − 12x 3 + 48x 2 − 64x

(0, 0) 1

2

−5

(4, 0) 5 Point of inflection

− 10 − 15

(2, − 16) Point of inflection

− 20 − 25 − 30

(1, − 27) Relative minimum

FIGURE 3.56

x

Original function

Begin by finding the intercepts of the graph. This function factors as

f 共x兲 ⫽ x共x3 ⫺ 12x2 ⫹ 48x ⫺ 64兲 ⫽ x共x ⫺ 4兲3.

Factored form

So, the x-intercepts occur when x ⫽ 0 and x ⫽ 4. The derivative is f⬘共x兲 ⫽ 4x3 ⫺ 36x2 ⫹ 96x ⫺ 64 ⫽ 4共x ⫺ 1兲共x ⫺ 4兲2.

First derivative Factored form

So, the critical numbers of f are x ⫽ 1 and x ⫽ 4. The second derivative of f is f ⬙ 共x兲 ⫽ 12x2 ⫺ 72x ⫹ 96 ⫽ 12共x ⫺ 4兲共x ⫺ 2兲

Second derivative Factored form

which implies that the second derivative is zero when x ⫽ 2 and x ⫽ 4. By testing the values of f⬘共x兲 and f ⬙ 共x兲, as shown in the table, you can see that f has one relative minimum and two points of inflection. The graph is shown in Figure 3.56. f 共x兲 x in 共⫺ ⬁, 1兲 x⫽1

⫺27

x in 共1, 2兲 x⫽2

⫺16

x in 共2, 4兲 x⫽4

0

x in 共4, ⬁兲

f⬘ 共x兲

f ⬙ 共x兲





Decreasing, concave upward

0



Relative minimum





Increasing, concave upward



0

Point of inflection





Increasing, concave downward

0

0

Point of inflection





Increasing, concave upward

Characteristics of graph

✓CHECKPOINT 2 Analyze the graph of f 共x兲 ⫽ x 4 ⫺ 4x3 ⫹ 5.



D I S C O V E RY A polynomial function of degree n can have at most n ⫺ 1 relative extrema and at most n ⫺ 2 points of inflection. For instance, the third-degree polynomial in Example 1 has two relative extrema and one point of inflection. Similarly, the fourth-degree polynomial function in Example 2 has one relative extremum and two points of inflection. Is it possible for a third-degree function to have no relative extrema? Is it possible for a fourth-degree function to have no relative extrema?

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 3.7

Example 3

D I S C O V E RY Show that the function in Example 3 can be rewritten as

f 共x兲 ⫽

共x ⫺ 2兲共2x ⫺ 2兲 ⫺ 共x2 ⫺ 2x ⫹ 4兲 共x ⫺ 2兲2 x共x ⫺ 4兲 . ⫽ 共x ⫺ 2兲2

x2 ⫺ x ⫺ 1 . x⫺1

共x ⫺ 2兲2共2x ⫺ 4兲 ⫺ 共x2 ⫺ 4x兲共2兲共x ⫺ 2兲 共x ⫺ 2兲4 2 共x ⫺ 2兲共2x ⫺ 8x ⫹ 8 ⫺ 2x2 ⫹ 8x兲 ⫽ 共x ⫺ 2兲4 8 . ⫽ 共x ⫺ 2兲3

f ⬙ 共x兲 ⫽

Vertical asymptote

−2

(0, − 2)

(4, 6) Relative minimum

4

6

Relative maximum

x 2 − 2x + 4 x−2

FIGURE 3.57

Factored form

x

Second derivative

Factored form

Because the second derivative has no zeros and because x ⫽ 2 is not in the domain of the function, you can conclude that the graph has no points of inflection. By testing the values of f⬘共x兲 and f ⬙ 共x兲, as shown in the table, you can see that f has one relative minimum and one relative maximum. The graph of f is shown in Figure 3.57.

−4

f (x) =

First derivative

So, the critical numbers of f are x ⫽ 0 and x ⫽ 4. The second derivative is

8

−4

Original function

f⬘共x兲 ⫽

y

2

x2 ⫺ 2x ⫹ 4 . x⫺2

The y-intercept occurs at 共0, ⫺2兲. Using the Quadratic Formula on the numerator, you can see that there are no x-intercepts. Because the denominator is zero when x ⫽ 2 (and the numerator is not zero when x ⫽ 2), it follows that x ⫽ 2 is a vertical asymptote of the graph. There are no horizontal asymptotes because the degree of the numerator is greater than the degree of the denominator. The derivative is

Use a graphing utility to graph f together with the line y ⫽ x. How do the two graphs compare as you zoom out? Describe what is meant by a “slant asymptote.” Find the slant asymptote of the

4

Analyzing a Graph

SOLUTION

4 . ⫽x⫹ x⫺2

6

269

Analyze the graph of

x2 ⫺ 2x ⫹ 4 f 共x兲 ⫽ x⫺2

function g共x兲 ⫽

Curve Sketching: A Summary

f 共x兲 x in 共⫺ ⬁, 0兲 x⫽0

⫺2

x in 共0, 2兲 x⫽2

f ⬙ 共x兲





Increasing, concave downward

0



Relative maximum





Decreasing, concave downward

Undef. Undef. Undef.

x in 共2, 4兲 x⫽4

f⬘ 共x兲

6

x in 共4, ⬁兲

Characteristics of graph

Vertical asymptote





Decreasing, concave upward

0



Relative minimum





Increasing, concave upward

✓CHECKPOINT 3 Analyze the graph of f 共x兲 ⫽

x2 . x⫺1



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270

CHAPTER 3

Applications of the Derivative

Example 4

Analyzing a Graph

Analyze the graph of f 共x兲 ⫽

2共x2 ⫺ 9兲 . x2 ⫺ 4

Begin by writing the function in factored form.

SOLUTION

f 共x兲 ⫽

Original function

2共x ⫺ 3兲共x ⫹ 3兲 共x ⫺ 2兲共x ⫹ 2兲

Factored form

The y-intercept is 共0, 92 兲, and the x-intercepts are 共⫺3, 0兲 and 共3, 0兲. The graph of f has vertical asymptotes at x ⫽ ± 2 and a horizontal asymptote at y ⫽ 2. The first derivative is

f (x) = y

f⬘共x兲 ⫽

2(x 2 − 9) x2 − 4



2关共x2 ⫺ 4兲共2x兲 ⫺ 共x2 ⫺ 9兲共2x兲兴 共x2 ⫺ 4兲2

First derivative

20x . 共x2 ⫺ 4兲2

Factored form

So, the critical number of f is x ⫽ 0. The second derivative of f is

共x2 ⫺ 4兲2共20兲 ⫺ 共20x兲共2兲共2x兲共x2 ⫺ 4兲 共x2 ⫺ 4兲4 2 2 20共x ⫺ 4兲共x ⫺ 4 ⫺ 4x2兲 ⫽ 共x2 ⫺ 4兲4 2 20共3x ⫹ 4兲 . ⫽⫺ 2 共x ⫺ 4兲3

f ⬙ 共x兲 ⫽ 4

−8

−4

(−3, 0)

FIGURE 3.58

( 0, 92 ) Relative minimum 4

(3, 0)

8

x

Second derivative

Factored form

Because the second derivative has no zeros and x ⫽ ± 2 are not in the domain of the function, you can conclude that the graph has no points of inflection. By testing the values of f⬘共x兲 and f ⬙ 共x兲, as shown in the table, you can see that f has one relative minimum. The graph of f is shown in Figure 3.58. f 共x兲 x in 共⫺ ⬁, ⫺2兲 x ⫽ ⫺2

f ⬙ 共x兲





Undef. Undef. Undef.

x in 共⫺2, 0兲 9 2

x⫽0 x in 共0, 2兲 x⫽2

f⬘ 共x兲

Decreasing, concave downward Vertical asymptote





Decreasing, concave upward

0



Relative minimum





Increasing, concave upward

Undef. Undef. Undef.

x in 共2, ⬁兲

Characteristics of graph





Vertical asymptote Increasing, concave downward

✓CHECKPOINT 4 Analyze the graph of f 共x兲 ⫽

x2 ⫹ 1 . x2 ⫺ 1



Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 3.7

Example 5

Curve Sketching: A Summary

271

Analyzing a Graph

Analyze the graph of

TECHNOLOGY

f 共x兲 ⫽ 2x5兾3 ⫺ 5x 4兾3.

Some graphing utilities will not graph the function in Example 5 properly if the function is entered as

SOLUTION

Original function

Begin by writing the function in factored form.

f 共x兲 ⫽ x 4兾3共2x1兾3 ⫺ 5兲

f 共x兲 ⫽ 2x^共5兾3兲 ⫺ 5x^共4兾3兲.

Factored form

One of the intercepts is 共0, 0兲. A second x-intercept occurs when 2x1兾3 ⫽ 5. 2x1兾3 ⫽ 5 x1兾3 ⫽ 52

To correct for this, you can enter the function as 3 x ^ 3 x ^ f 共x兲 ⫽ 2共冪 兲 5 ⫺ 5共冪 兲 4.

x ⫽ 共52 兲

3

x ⫽ 125 8

Try entering both functions into a graphing utility to see whether both functions produce correct graphs.

The first derivative is f⬘共x兲 ⫽ ⫽

10 2兾3 1兾3 ⫺ 20 3 x 3 x 10 1兾3 1兾3 ⫺ 2兲. 3 x 共x

First derivative Factored form

So, the critical numbers of f are x ⫽ 0 and x ⫽ 8. The second derivative is

Algebra Review For help on the algebra in Example 5, see Example 2(a) in the Chapter 3 Algebra Review, on page 284.

f ⬙ 共x兲 ⫽ ⫽ ⫽

20 ⫺1兾3 ⫺2兾3 ⫺ 20 9 x 9 x 20 ⫺2兾3 1兾3 共x ⫺ 1兲 9 x

20共x1兾3 ⫺ 1兲 . 9x2兾3

Second derivative

Factored form

So, possible points of inflection occur when x ⫽ 1 and when x ⫽ 0. By testing the values of f⬘共x兲 and f ⬙ 共x兲, as shown in the table, you can see that f has one relative maximum, one relative minimum, and one point of inflection. The graph of f is shown in Figure 3.59. y

f (x) = 2x 5/3 − 5x 4/3 Relative maximum (0, 0) 4

−4

( 1258 , 0) 8

(1, −3) Point of inflection

12

f 共x兲 x

f⬘ 共x兲

f ⬙ 共x兲





0

Undef.





Decreasing, concave downward



0

Point of inflection





Decreasing, concave upward

0



Relative minimum





Increasing, concave upward

x in 共⫺ ⬁, 0兲 x⫽0

0

x in 共0, 1兲 x⫽1

⫺3

x in 共1, 8兲 x⫽8 (8, − 16) Relative minimum

⫺16

x in 共8, ⬁兲

Characteristics of graph Increasing, concave downward Relative maximum

FIGURE 3.59

✓CHECKPOINT 5 Analyze the graph of f 共x兲 ⫽ 2x3兾2 ⫺ 6x1兾2.



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272

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Summary of Simple Polynomial Graphs A summary of the graphs of polynomial functions of degrees 0, 1, 2, and 3 is shown in Figure 3.60. Because of their simplicity, lower-degree polynomial functions are commonly used as mathematical models. Constant function (degree 0):

Linear function (degree 1):

y=a

y = ax + b

Line of slope a

Horizontal line

a Quadratic function (degree 2): y = ax + bx + c

y = ax 3 + bx 2 + cx + d

Parabola

Cubic curve

0

a

0

Cubic function (degree 3):

2

a

a

0

0

a

0

a

0

FIGURE 3.60

STUDY TIP The graph of any cubic polynomial has one point of inflection. The slope of the graph at the point of inflection may be zero or nonzero.

CONCEPT CHECK 1. A fourth-degree polynomial can have at most how many relative extrema? 2. A fourth-degree polynomial can have at most how many points of inflection? 3. Complete the following: A polynomial function of degree n can have at most ______ relative extrema. 4. Complete the following: A polynomial function of degree n can have at most ______ points of inflection.

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SECTION 3.7

Curve Sketching: A Summary

273

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 3.1 and 3.6.

Skills Review 3.7

In Exercises 1– 4, find the vertical and horizontal asymptotes of the graph. 1. f 共x兲 ⫽

1 x2

2. f 共x兲 ⫽

8 共x ⫺ 2兲2

3. f 共x兲 ⫽

40x x⫹3

4. f 共x兲 ⫽

x2 ⫺ 3 x2 ⫺ 4x ⫹ 3

In Exercises 5–10, determine the open intervals on which the function is increasing or decreasing. 5. f 共x兲 ⫽ x2 ⫹ 4x ⫹ 2 8. f 共x兲 ⫽

⫺x3

6. f 共x兲 ⫽ ⫺x2 ⫺ 8x ⫹ 1

x2

⫹ ⫺1 x2

9. f 共x兲 ⫽

x⫺2 x⫺1

Exercises 3.7

1. y ⫽ ⫺x2 ⫺ 2x ⫹ 3

2. y ⫽ 2x2 ⫺ 4x ⫹ 1

3. y ⫽ x3 ⫺ 4x2 ⫹ 6

4. y ⫽ ⫺x3 ⫹ x ⫺ 2

5. y ⫽ 2 ⫺ x ⫺ x3

6. y ⫽ x3 ⫹ 3x2 ⫹ 3x ⫹ 2

7. y ⫽ 3x3 ⫺ 9x ⫹ 1

8. y ⫽ ⫺4x3 ⫹ 6x2 10. y ⫽ x 4 ⫺ 2x2

11. y ⫽ x3 ⫺ 6x2 ⫹ 3x ⫹ 10 12. y ⫽ ⫺x3 ⫹ 3x2 ⫹ 9x ⫺ 2 13. y ⫽ x 4 ⫺ 8x3 ⫹ 18x2 ⫺ 16x ⫹ 5 15. y ⫽ x 4 ⫺ 4x3 ⫹ 16x

16. y ⫽ x5 ⫹ 1

17. y ⫽ x5 ⫺ 5x

18. y ⫽ 共x ⫺ 1兲5

19. y ⫽ 21. y ⫽

⫹1 x



x2 ⫹ 1, x ≤ 0 1 ⫺ 2x, x > 0

20.

y⫽

22. y ⫽

x⫹2 x



x2 ⫹ 4, x < 0 4 ⫺ x, x ≥ 0

In Exercises 23–34, use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. 23. y ⫽

x2

x2 ⫹3

25. y ⫽ 3x2兾3 ⫺ 2x 27. y ⫽ 1 ⫺

x2兾3

29. y ⫽ x1兾3 ⫹ 1

24. y ⫽

31. y ⫽ x5兾3 ⫺ 5x2兾3

32. y ⫽ x 4兾3

33. y ⫽ x冪x 2 ⫺ 9

34. y ⫽

x 冪x 2 ⫺ 4

In Exercises 35– 44, sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. 35. y ⫽

5 ⫺ 3x x⫺2

36. y ⫽

x2 ⫹ 1 x2 ⫺ 9

37. y ⫽

2x x ⫺1

38. y ⫽

x2 ⫺ 6x ⫹ 12 x⫺4

2

39. y ⫽ x冪4 ⫺ x

14. y ⫽ x 4 ⫺ 4x3 ⫹ 16x ⫺ 16

x2

10. f 共x兲 ⫽ ⫺x3 ⫺ 4x2 ⫹ 3x ⫹ 2

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–22, sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph.

9. y ⫽ 3x 4 ⫹ 4x3

7. f 共x兲 ⫽ x3 ⫺ 3x ⫹ 1

40. y ⫽ x冪4 ⫺ x2

41. y ⫽

x⫺3 x

42. y ⫽ x ⫹

43. y ⫽

x3 x ⫺1

44. y ⫽

3

32 x2

x4 x ⫺1 4

In Exercises 45 and 46, find values of a, b, c, and d such that the graph of f 冇x冈 ⴝ ax 3 1 bx2 1 cx 1 d will resemble the given graph. Then use a graphing utility to verify your result. (There are many correct answers.) 45.

46.

x x2 ⫹ 1

26. y ⫽ 3x2兾3 ⫺ x2 28. y ⫽ 共1 ⫺ x兲2兾3 30. y ⫽ x⫺1兾3

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274

CHAPTER 3

Applications of the Derivative

In Exercises 47–50, use the graph of f⬘ or f⬙ to sketch the graph of f. (There are many correct answers.) y

47.

y

48.

3

f

2

3

4

−1

1

2 y

50.

4

2

3

1

f

f

−2 −1

1 1

(a) Find the cost C as a function of s for a 100-mile trip on an interstate highway. 1

2

x

x

2

In Exercises 51 and 52, sketch a graph of a function f having the given characteristics. (There are many correct answers.) 51. f 共⫺2兲 ⫽ f 共0兲 ⫽ 0

52. f 共⫺1兲 ⫽ f 共3兲 ⫽ 0

f⬘共x兲 > 0 if x < ⫺1

f⬘共1兲 is undefined.

f⬘共x兲 < 0 if ⫺1 < x < 0

f⬘共x兲 < 0 if x < 1

f⬘共x兲 > 0 if x > 0

f⬘共x兲 > 0 if x > 1

f⬘共⫺1兲 ⫽ f⬘共0兲 ⫽ 0

f ⬙ 共x兲 < 0, x ⫽ 1 lim f 共x兲 ⫽ 4

x→ ⬁

In Exercises 53 and 54, create a function whose graph has the given characteristics. (There are many correct answers.) 53. Vertical asymptote: x ⫽ 5 54. Vertical asymptote: x ⫽ ⫺3 Horizontal asymptote: None 55. MAKE A DECISION: SOCIAL SECURITY The table lists the average monthly Social Security benefits B (in dollars) for retired workers aged 62 and over from 1998 through 2005. A model for the data is 582.6 ⫹ 38.38t , 1 ⫹ 0.025t ⫺ 0.0009t2

(b) Use a graphing utility to graph the function found in part (a) and determine the most economical speed. 57. MAKE A DECISION: PROFIT The management of a company is considering three possible models for predicting the company’s profits from 2003 through 2008. Model I gives the expected annual profits if the current trends continue. Models II and III give the expected annual profits for various combinations of increased labor and energy costs. In each model, p is the profit (in billions of dollars) and t ⫽ 0 corresponds to 2003. Model I:

p ⫽ 0.03t 2 ⫺ 0.01t ⫹ 3.39

Model II: p ⫽ 0.08t ⫹ 3.36 Model III: p ⫽ ⫺0.07t 2 ⫹ 0.05t ⫹ 3.38 (a) Use a graphing utility to graph all three models in the same viewing window. (b) For which models are profits increasing during the interval from 2003 through 2008? (c) Which model is the most optimistic? Which is the most pessimistic? Which model would you choose? Explain. 58. Meteorology The monthly normal temperature T (in degrees Fahrenheit) for Pittsburgh, Pennsylvania can be modeled by

Horizontal asymptote: y ⫽ 0

B⫽

(c) Should this model be used to predict the average monthly Social Security benefits in future years? Why or why not? 56. Cost An employee of a delivery company earns $10 per hour driving a delivery van in an area where gasoline costs $2.80 per gallon. When the van is driven at a constant speed s (in miles per hour, with 40 ≤ s ≤ 65), the van gets 700兾s miles per gallon.

−3

y

−2 −1

x

f

−2 x

49.

1

−1

1 −2

(b) Use the model to predict the average monthly benefit in 2008.

1

4

(a) Use a graphing utility to create a scatter plot of the data and graph the model in the same viewing window. How well does the model fit the data?

8 ≤ t ≤ 15

where t ⫽ 8 corresponds to 1998. Security Administration)

(Source: U.S. Social

22.329 ⫺ 0.7t ⫹ 0.029t 2 , 1 ≤ t ≤ 12 1 ⫺ 0.203t ⫹ 0.014t 2 where t is the month, with t ⫽ 1 corresponding to January. Use a graphing utility to graph the model and find all absolute extrema. Interpret the meaning of these values in the context of the problem. (Source: National Climatic Data Center) T⫽

t

8

9

10

11

12

13

14

15

Writing In Exercises 59 and 60, use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function may indicate that there should be one.

B

780

804

844

874

895

922

955

1002

59. h共x兲 ⫽

6 ⫺ 2x 3⫺x

60. g共x兲 ⫽

x2 ⫹ x ⫺ 2 x⫺1

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SECTION 3.8

Differentials and Marginal Analysis

275

Section 3.8

Differentials and Marginal Analysis

■ Find the differentials of functions. ■ Use differentials to approximate changes in functions. ■ Use differentials to approximate changes in real-life models.

Differentials When the derivative was defined in Section 2.1 as the limit of the ratio ⌬y兾⌬x, it seemed natural to retain the quotient symbolism for the limit itself. So, the derivative of y with respect to x was denoted by dy ⌬y ⫽ lim dx ⌬x→0 ⌬x even though we did not interpret dy兾dx as the quotient of two separate quantities. In this section, you will see that the quantities dy and dx can be assigned meanings in such a way that their quotient, when dx ⫽ 0, is equal to the derivative of y with respect to x. STUDY TIP In this definition, dx can have any nonzero value. In most applications, however, dx is chosen to be small and this choice is denoted by dx ⫽ ⌬x.

Definition of Differentials

Let y ⫽ f 共x兲 represent a differentiable function. The differential of x (denoted by dx) is any nonzero real number. The differential of y (denoted by dy) is dy ⫽ f⬘共x兲 dx. One use of differentials is in approximating the change in f 共x兲 that corresponds to a change in x, as shown in Figure 3.61. This change is denoted by ⌬y ⫽ f 共x ⫹ ⌬x兲 ⫺ f 共x兲.

STUDY TIP Note in Figure 3.61 that near the point of tangency, the graph of f is very close to the tangent line. This is the essence of the approximations used in this section. In other words, near the point of tangency, dy ⬇ ⌬y.

Change in y

In Figure 3.61, notice that as ⌬x gets smaller and smaller, the values of dy and ⌬y get closer and closer. That is, when ⌬x is small, dy ⬇ ⌬y. y

(x

Δx, f (x

Δx)) Δy

dy

(x, f (x))

dx

x

Δx

x

Δx

x

FIGURE 3.61

This tangent line approximation is the basis for most applications of differentials.

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276

CHAPTER 3

Applications of the Derivative

Example 1

Interpreting Differentials Graphically

Consider the function given by f 共x兲 ⫽ x2.

Find the value of dy when x ⫽ 1 and dx ⫽ 0.01. Compare this with the value of ⌬y when x ⫽ 1 and ⌬x ⫽ 0.01. Interpret the results graphically.

y = 2x − 1

SOLUTION f(1.01)

f(x) = x 2

dy

(1, 1)

Δx

Original function

Begin by finding the derivative of f.

f⬘共x兲 ⫽ 2x

Derivative of f

When x ⫽ 1 and dx ⫽ 0.01, the value of the differential dy is dy ⫽ f⬘共x兲 dx ⫽ f⬘共1兲共0.01兲 ⫽ 2共1兲共0.01兲 ⫽ 0.02.

Δy

Differential of y Substitute 1 for x and 0.01 for dx. Use f⬘共x兲 ⫽ 2x. Simplify.

When x ⫽ 1 and ⌬x ⫽ 0.01, the value of ⌬y is

f(1)

0.01

FIGURE 3.62

⌬y ⫽ f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⫽ f 共1.01兲 ⫺ f 共1兲 ⫽ 共1.01兲2 ⫺ 共1兲2 ⫽ 0.0201.

Change in y Substitute 1 for x and 0.01 for ⌬x.

Simplify.

Note that dy ⬇ ⌬y, as shown in Figure 3.62.

✓CHECKPOINT 1 Find the value of dy when x ⫽ 2 and dx ⫽ 0.01 for f 共x) ⫽ x 4. Compare this with the value of ⌬y when x ⫽ 2 and ⌬x ⫽ 0.01. ■ STUDY TIP Find an equation of the tangent line y ⫽ g共x兲 to the graph of f 共x) ⫽ x2 at the point x ⫽ 1. Evaluate g共1.01兲 and f 共1.01兲.

The validity of the approximation dy ⬇ ⌬y, dx ⫽ 0 stems from the definition of the derivative. That is, the existence of the limit f⬘共x兲 ⫽ lim

⌬x→0

f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x

implies that when ⌬x is close to zero, then f⬘共x兲 is close to the difference quotient. So, you can write f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⬇ f⬘共x兲 ⌬x f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⬇ f⬘共x兲 ⌬x ⌬y ⬇ f⬘共x兲 ⌬x. Substituting dx for ⌬x and dy for f⬘共x兲 dx produces ⌬y ⬇ dy.

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SECTION 3.8

Differentials and Marginal Analysis

277

Marginal Analysis Differentials are used in economics to approximate changes in revenue, cost, and profit. Suppose that R ⫽ f 共x兲 is the total revenue for selling x units of a product. When the number of units increases by 1, the change in x is ⌬x ⫽ 1, and the change in R is ⌬R ⫽ f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⬇ dR ⫽

dR dx. dx

In other words, you can use the differential dR to approximate the change in the revenue that accompanies the sale of one additional unit. Similarly, the differentials dC and dP can be used to approximate the changes in cost and profit that accompany the sale (or production) of one additional unit.

Example 2

Using Marginal Analysis

The demand function for a product is modeled by p ⫽ 400 ⫺ x,

0 ≤ x ≤ 400.

Use differentials to approximate the change in revenue as sales increase from 149 units to 150 units. Compare this with the actual change in revenue. SOLUTION

Begin by finding the marginal revenue, dR兾dx. R ⫽ xp ⫽ x共400 ⫺ x兲

Formula for revenue Use p ⫽ 400 ⫺ x

Multiply. ⫽ 400x ⫺ x dR ⫽ 400 ⫺ 2x Power Rule dx When x ⫽ 149 and dx ⫽ ⌬x ⫽ 1, you can approximate the change in the revenue to be 2

关400 ⫺ 2共149兲兴共1兲 ⫽ $102. When x increases from 149 to 150, the actual change in revenue is ⌬R ⫽ 关400共150兲 ⫺ 1502兴 ⫺ 关400共149兲 ⫺ 1492兴 ⫽ 37,500 ⫺ 37,399 ⫽ $101

✓CHECKPOINT 2 The demand function for a product is modeled by p ⫽ 200 ⫺ x,

0 ≤ x ≤ 200.

Use differentials to approximate the change in revenue as sales increase from 89 to 90 units. Compare this with the actual change in revenue. ■

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278

CHAPTER 3

Applications of the Derivative

Example 3 MAKE A DECISION

Using Marginal Analysis

The profit derived from selling x units of an item is modeled by P ⫽ 0.0002x3 ⫹ 10x. Use the differential dP to approximate the change in profit when the production level changes from 50 to 51 units. Compare this with the actual gain in profit obtained by increasing the production level from 50 to 51 units. Will the gain in profit exceed $11? STUDY TIP Example 3 uses differentials to solve the same problem that was solved in Example 5 in Section 2.3. Look back at that solution. Which approach do you prefer?

SOLUTION

The marginal profit is

dP ⫽ 0.0006x2 ⫹ 10. dx When x ⫽ 50 and dx ⫽ 1, the differential is

关0.0006共50兲2 ⫹ 10兴共1兲 ⫽ $11.50. When x changes from 50 to 51 units, the actual change in profit is ⌬P ⫽ 关共0.0002兲共51兲3 ⫹ 10共51兲兴 ⫺ 关共0.0002兲共50兲3 ⫹ 10共50兲兴 ⬇ 536.53 ⫺ 525.00 ⫽ $11.53. These values are shown graphically in Figure 3.63. Note that the gain in profit will exceed $11. Marginal Profit P 600

(51, 536.53) dP ≈ ΔP

Profit (in dollars)

500

STUDY TIP Find an equation of the tangent line y ⫽ f 共x兲 to the graph of P ⫽ 0.0002x3 ⫹ 10x at the point x ⫽ 50. Evaluate f 共51兲 and p共51兲.

400 300

dP ΔP

(50, 525) Δx = dx ΔP = $11.53 dP = $11.50

200 100

P = 0.0002x 3 + 10x 10

20

30

40

50

x

Number of units

FIGURE 3.63

✓CHECKPOINT 3 Use the differential dP to approximate the change in profit for the profit function in Example 3 when the production level changes from 40 to 41 units. Compare this with the actual gain in profit obtained by increasing the production level from 40 to 41 units. ■

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SECTION 3.8

Differentials and Marginal Analysis

279

Formulas for Differentials You can use the definition of differentials to rewrite each differentiation rule in differential form. For example, if u and v are differentiable functions of x, then du ⫽ 共du兾dx兲 dx and dv ⫽ 共dv兾dx兲 dx, which implies that you can write the Product Rule in the following differential form. d 关uv兴 ⫽

d 关uv兴 dx dx

Differential of uv

dv du dx ⫹v dx dx





⫽ u

Product Rule

dv du dx ⫹ v dx dx dx ⫽ u dv ⫹ v du ⫽u

Differential form of Product Rule

The following summary gives the differential forms of the differentiation rules presented so far in the text. Differential Forms of Differentiation Rules

Constant Multiple Rule:

d 关cu兴 ⫽ c du

Sum or Difference Rule:

d 关u ± v兴 ⫽ du ± dv

Product Rule:

d 关uv兴 ⫽ u dv ⫹ v du

Quotient Rule:

d

Constant Rule:

d 关c兴 ⫽ 0

Power Rule:

d 关x n兴 ⫽ nx n⫺1 dx

冤 uv冥 ⫽ v du v⫺ u dv 2

The next example compares the derivatives and differentials of several simple functions.

Example 4

Finding Differentials

Find the differential dy of each function.

✓CHECKPOINT 4

Function

Find the differential dy of each function.

a. y ⫽ x2

a. y ⫽ 4x3

b. y ⫽

b. y ⫽

2x ⫹ 1 3

c. y ⫽

3x2

1 d. y ⫽ 2 x

c. y ⫽ 2x2 ⫺ 3x

⫺ 2x d. y ⫽ ■

3x ⫹ 2 5

1 x

Derivative

Differential

dy ⫽ 2x dx

dy ⫽ 2x dx

dy 3 ⫽ dx 5

dy ⫽

dy ⫽ 4x ⫺ 3 dx

dy ⫽ 共4x ⫺ 3兲 dx

dy 1 ⫽⫺ 2 dx x

dy ⫽ ⫺

3 dx 5

1 dx x2

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280

CHAPTER 3

Applications of the Derivative

Error Propagation A common use of differentials is the estimation of errors that result from inaccuracies of physical measuring devices. This is shown in Example 5.

Example 5

Estimating Measurement Errors

The radius of a ball bearing is measured to be 0.7 inch, as shown in Figure 3.64. This implies that the volume of the ball bearing is 43␲ 共0.7兲3 ⬇ 1.4368 cubic inches. You are told that the measurement of the radius is correct to within 0.01 inch. How far off could the calculation of the volume be? SOLUTION

Because the value of r can be off by 0.01 inch, it follows that

⫺0.01 ≤ ⌬r ≤ 0.01.

Possible error in measuring

Using ⌬r ⫽ dr, you can estimate the possible error in the volume. V ⫽ 43␲ r 3

0.7 in.

dV ⫽ FIGURE 3.64

Formula for differential of V

The possible error in the volume is 4␲ r 2 dr ⫽ 4␲ 共0.7兲2共± 0.01兲 ⬇ ± 0.0616 cubic inch.

✓CHECKPOINT 5 Find the surface area of the ball bearing in Example 5. How far off could your calculation of the surface area be? The surface area of a sphere is given by S ⫽ 4␲ r 2. ■

dV dr ⫽ 4␲r2 dr dr

Formula for volume

Substitute for r and dr. Possible error

So, the volume of the ball bearing could range between

共1.4368 ⫺ 0.0616兲 ⫽1.3752 cubic inches and

共1.4368 ⫹ 0.0616兲 ⫽ 1.4984 cubic inches. In Example 5, the relative error in the volume is defined to be the ratio of dV to V. This ratio is dV ± 0.0616 ⬇ ⬇ ± 0.0429. V 1.4368 This corresponds to a percentage error of 4.29%.

CONCEPT CHECK 1. Given a differentiable function y ⴝ f 冇x冈, what is the differential of x? 2. Given a differentiable function y ⴝ f 冇x冈, write an expression for the differential of y. 3. Write the differential form of the Quotient Rule. 4. When using differentials, what is meant by the terms relative error and percentage error?

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SECTION 3.8

Skills Review 3.8

Differentials and Marginal Analysis

281

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 2.2 and 2.4.

In Exercises 1–12, find the derivative. 1. C ⫽ 44 ⫹ 0.09x2

2. C ⫽ 250 ⫹ 0.15x

3. R ⫽ x共1.25 ⫹ 0.02冪x 兲

4. R ⫽ x共15.5 ⫺ 1.55x兲

5. P ⫽ ⫺0.03x1兾3 ⫹ 1.4x ⫺ 2250

6. P ⫽ ⫺0.02x 2 ⫹ 25x ⫺ 1000

8. A ⫽

9. C ⫽ 2␲ r

7. A ⫽

1 2 4 冪3 x

10. P ⫽ 4w

11. S ⫽

6x 2

12. P ⫽ 2x ⫹ 冪2 x

4␲ r 2

In Exercises 13–16, write a formula for the quantity. 13. Area A of a circle of radius r

14. Area A of a square of side x

15. Volume V of a cube of edge x

16. Volume V of a sphere of radius r

Exercises 3.8

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

15. y ⫽ x2

In Exercises 1– 6, find the differential dy. 1. y ⫽ 3x2 ⫺ 4

2. y ⫽ 3x2兾3

3. y ⫽ 共4x ⫺ 1兲3

4. y ⫽

5. y ⫽ 冪9 ⫺ x 2

3 6x2 6. y ⫽ 冪

x⫹1 2x ⫺ 1

In Exercises 7–10, let x ⴝ 1 and ⌬x ⴝ 0.01. Find ⌬y. 7. f 共x兲 ⫽ 5x2 ⫺ 1 9. f 共x兲 ⫽

8. f 共x兲 ⫽ 冪3x

4

10. f 共x兲 ⫽

3 x 冪

x2

x ⫹1

In Exercises 11–14, compare the values of dy and ⌬y. 11. y ⫽ 0.5x3

x⫽2

12. y ⫽ 1 ⫺ 2x2

x⫽0

⌬x ⫽ dx ⫽ 0.1 ⌬x ⫽ dx ⫽ ⫺0.1

13. y ⫽ x 4 ⫹ 1

x ⫽ ⫺1

⌬x ⫽ dx ⫽ 0.01

14. y ⫽ 2x ⫹ 1

x⫽2

⌬x ⫽ dx ⫽ 0.01

In Exercises 15–20, let x ⴝ 2 and complete the table for the function. dx ⫽ ⌬x 1.000

dy

⌬y

⌬y ⫺ dy

dy兾⌬y

17. y ⫽

1 x2

18. y ⫽

1 x

16. y ⫽ x5

4 x 19. y ⫽ 冪

20. y ⫽ 冪x In Exercises 21–24, find an equation of the tangent line to the function at the given point. Then find the function values and the tangent line values at f 冇x 1 ⌬x冈 and y 冇x 1 ⌬x冈 for ⌬x ⴝ ⴚ0.01 and 0.01. Function

Point

21. f 共x兲 ⫽ 2x ⫺ x ⫹ 1

共⫺2, ⫺19兲

22. f 共x兲 ⫽ 3x ⫺ 1 x 23. f 共x兲 ⫽ 2 x ⫹1

共2, 11兲

24. f 共x兲 ⫽ 冪25 ⫺ x2

共3, 4兲

3

2

2

25. Profit

共0, 0兲

The profit P for a company producing x units is

P ⫽ 共500x ⫺ x2兲 ⫺

冢12x

2



⫺ 77x ⫹ 3000 .

0.500

Approximate the change and percent change in profit as production changes from x ⫽ 115 to x ⫽ 120 units.

0.100

26. Revenue The revenue R for a company selling x units is

0.010

R ⫽ 900x ⫺ 0.1x2.

0.001

Use differentials to approximate the change in revenue if sales increase from x ⫽ 3000 to x ⫽ 3100 units.

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282

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Applications of the Derivative

27. Demand The demand function for a product is modeled by p ⫽ 75 ⫺ 0.25x. (a) If x changes from 7 to 8, what is the corresponding change in p? Compare the values of ⌬ p and dp. (b) Repeat part (a) when x changes from 70 to 71 units. 28. Biology: Wildlife Management A state game commission introduces 50 deer into newly acquired state game lands. The population N of the herd can be modeled by N⫽

10共5 ⫹ 3t兲 1 ⫹ 0.04t

where t is the time in years. Use differentials to approximate the change in the herd size from t ⫽ 5 to t ⫽ 6. Marginal Analysis In Exercises 29–34, use differentials to approximate the change in cost, revenue, or profit corresponding to an increase in sales of one unit. For instance, in Exercise 29, approximate the change in cost as x increases from 12 units to 13 units. Then use a graphing utility to graph the function, and use the trace feature to verify your result. Function

x-Value

29. C ⫽ 0.05x ⫹ 4x ⫹ 10

x ⫽ 12

30. C ⫽ 0.025x 2 ⫹ 8x ⫹ 5

x ⫽ 10

31. R ⫽ 30x ⫺ 0.15x

x ⫽ 75

32. R ⫽ 50x ⫺ 1.5x 2

x ⫽ 15

33. P ⫽ ⫺0.5x ⫹ 2500x ⫺ 6000

x ⫽ 50

34. P ⫽ ⫺x2 ⫹ 60x ⫺ 100

x ⫽ 25

2

2

3

35. Marginal Analysis A retailer has determined that the monthly sales x of a watch are 150 units when the price is $50, but decrease to 120 units when the price is $60. Assume that the demand is a linear function of the price. Find the revenue R as a function of x and approximate the change in revenue for a one-unit increase in sales when x ⫽ 141. Make a sketch showing dR and ⌬R. 36. Marginal Analysis A manufacturer determines that the demand x for a product is inversely proportional to the square of the price p. When the price is $10, the demand is 2500. Find the revenue R as a function of x and approximate the change in revenue for a one-unit increase in sales when x ⫽ 3000. Make a sketch showing dR and ⌬R. 37. Marginal Analysis The demand x for a web camera is 30,000 units per month when the price is $25 and 40,000 units when the price is $20. The initial investment is $275,000 and the cost per unit is $17. Assume that the demand is a linear function of the price. Find the profit P as a function of x and approximate the change in profit for a one-unit increase in sales when x ⫽ 28,000. Make a sketch showing dP and ⌬P.

38. Marginal Analysis The variable cost for the production of a calculator is $14.25 and the initial investment is $110,000. Find the total cost C as a function of x, the number of units produced. Then use differentials to approximate the change in the cost for a one-unit increase in production when x ⫽ 50,000. Make a sketch showing dC and ⌬C. Explain why dC ⫽ ⌬C in this problem. 39. Area The side of a square is measured to be 12 inches, 1 with a possible error of 64 inch. Use differentials to approximate the possible error and the relative error in computing the area of the square. 40. Volume The radius of a sphere is measured to be 6 inches, with a possible error of 0.02 inch. Use differentials to approximate the possible error and the relative error in computing the volume of the sphere. 41. Economics: Gross Domestic Product The gross domestic product (GDP) of the United States for 2001 through 2005 is modeled by G ⫽ 0.0026x2 ⫺ 7.246x ⫹ 14,597.85 where G is the GDP (in billions of dollars) and x is the capital outlay (in billions of dollars). Use differentials to approximate the change in the GDP when the capital outlays change from $2100 billion to $2300 billion. (Source: U.S. Bureau of Economic Analysis, U.S. Office of Management and Budget) 42. Medical Science The concentration C (in milligrams per milliliter) of a drug in a patient’s bloodstream t hours after injection into muscle tissue is modeled by C⫽

3t . 27 ⫹ t 3

Use differentials to approximate the change in the concentration when t changes from t ⫽ 1 to t ⫽ 1.5. 43. Physiology: Body Surface Area The body surface area (BSA) of a 180-centimeter-tall (about six-feet-tall) person is modeled by B ⫽ 0.1冪5w where B is the BSA (in square meters) and w is the weight (in kilograms). Use differentials to approximate the change in the person’s BSA when the person’s weight changes from 90 kilograms to 95 kilograms. True or False? In Exercises 44 and 45, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. 44. If y ⫽ x ⫹ c, then dy ⫽ dx. 45. If y ⫽ ax ⫹ b, then ⌬y兾⌬x ⫽ dy兾dx.

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Algebra Review

283

Algebra Review Solving Equations Much of the algebra in Chapter 3 involves simplifying algebraic expressions (see pages 196 and 197) and solving algebraic equations (see page 106). The Algebra Review on page 106 illustrates some of the basic techniques for solving equations. On these two pages, you can review some of the more complicated techniques for solving equations. When solving an equation, remember that your basic goal is to isolate the variable on one side of the equation. To do this, you use inverse operations. For instance, to get rid of the subtract 2 in x⫺2⫽0 you add 2 to each side of the equation. Similarly, to get rid of the square root in 冪x ⫹ 3 ⫽ 2

you square both sides of the equation.

Example 1

Solving an Equation

Solve each equation. a.

36共x 2 ⫺ 1兲 ⫽0 共x 2 ⫹ 3兲3

b. 0 ⫽ 2 ⫺

288 x2

c. 0 ⫽ 2x共2x 2 ⫺ 3兲

SOLUTION

a.

36共x2 ⫺ 1兲 ⫽0 共x2 ⫹ 3兲3

Example 2, page 227

36共x2 ⫺ 1兲 ⫽ 0

A fraction is zero only if its numerator is zero.

x2

⫺1⫽0 x2

Divide each side by 36.

⫽1

Add 1 to each side.

x ⫽ ±1 b.

0⫽2⫺ ⫺2 ⫽ ⫺

Take the square root of each side.

288 x2

Example 2, page 237

288 x2

Subtract 2 from each side.

144 x2

Divide each side by ⫺2.

x2 ⫽ 144

Multiply each side by x2.

1⫽

x ⫽ ± 12 c.

Take the square root of each side.

0 ⫽ 2x共

2x2

2x ⫽ 0 2x2 ⫺ 3 ⫽ 0

⫺ 3兲

Example 3, page 238

x⫽0

Set first factor equal to zero.

x ⫽ ±冪

3 2

Set second factor equal to zero.

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284

CHAPTER 3

Example 2

Applications of the Derivative

Solve an Equation

Solve each equation. a.

20共x 1兾3 ⫺ 1兲 ⫽0 9x 2兾3

c. x 2共4x ⫺ 3兲 ⫽ 0

b.

d.

25 冪x

⫺ 0.5 ⫽ 0

4x ⫽0 3共x 2 ⫺ 4兲1兾3

e. g⬘共x兲 ⫽ 0, where g共x兲 ⫽ 共x ⫺ 2兲共x ⫹ 1兲2 SOLUTION

a.

20共x1兾3 ⫺ 1兲 ⫽0 9x2兾3

Example 5, page 271

20共x1兾3 ⫺ 1兲 ⫽ 0

A fraction is zero only if its numerator is zero.

x1兾3 ⫺ 1 ⫽ 0 x

1兾3

Divide each side by 20.

⫽1

Add 1 to each side.

x⫽1 b.

25 冪x

Cube each side.

⫺ 0.5 ⫽ 0 25 冪x

Example 4, page 248

⫽ 0.5

Add 0.5 to each side.

25 ⫽ 0.5冪x

Multiply each side by 冪x.

50 ⫽ 冪x

Divide each side by 0.5.

2500 ⫽ x

Square both sides.

c. x2共4x ⫺ 3兲 ⫽ 0

Example 2, page 218

x ⫽0

x⫽0

4x ⫺ 3 ⫽ 0

3 4

2

d.

x⫽

Set first factor equal to zero. Set second factor equal to zero.

4x ⫽0 3共x2 ⫺ 4兲1兾3

Example 4, page 210

4x ⫽ 0

A fraction is zero only if its numerator is zero.

x⫽0

Divide each side by 4.

e. g共x兲 ⫽ 共x ⫺ 2兲(x ⫹ 1兲2

Exercise 45, page 233

共x ⫺ 2兲共2兲共x ⫹ 1兲 ⫹ 共x ⫹ 1兲 共1兲 ⫽ 0 2

Find derivative and set equal to zero.

共x ⫹ 1兲关2共x ⫺ 2兲 ⫹ 共x ⫹ 1兲兴 ⫽ 0

Factor.

共x ⫹ 1兲共2x ⫺ 4 ⫹ x ⫹ 1兲 ⫽ 0

Multiply factors.

共x ⫹ 1兲共3x ⫺ 3兲 ⫽ 0 x⫹1⫽0 3x ⫺ 3 ⫽ 0

Combine like terms.

x ⫽ ⫺1

Set first factor equal to zero.

x⫽1

Set second factor equal to zero.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Chapter Summary and Study Strategies

Chapter Summary and Study Strategies After studying this chapter, you should have acquired the following skills. The exercise numbers are keyed to the Review Exercises that begin on page 287. Answers to odd-numbered Review Exercises are given in the back of the text.*

Section 3.1 ■

Find the critical numbers of a function.



Find the open intervals on which a function is increasing or decreasing.

Review Exercises 1– 4

c is a critical number of f if f⬘共c兲 ⫽ 0 or f⬘共c兲 is undefined. 5– 8

Increasing if f⬘共x兲 > 0 Decreasing if f⬘共x兲 < 0 ■

Find intervals on which a real-life model is increasing or decreasing, and interpret the results in context.

9, 10, 95

Section 3.2 ■

Use the First-Derivative Test to find the relative extrema of a function.

11–20



Find the absolute extrema of a continuous function on a closed interval.

21–30



Find minimum and maximum values of a real-life model and interpret the results in context.

31, 32

Section 3.3 ■

Find the open intervals on which the graph of a function is concave upward or concave downward.

33–36

Concave upward if f ⬙ 共x兲 > 0 Concave downward if f ⬙ 共x兲 < 0 ■

Find the points of inflection of the graph of a function.

37– 40



Use the Second-Derivative Test to find the relative extrema of a function.

41– 44



Find the point of diminishing returns of an input-output model.

45, 46

Section 3.4 ■

Solve real-life optimization problems.

47–53, 96

Section 3.5 ■

Solve business and economics optimization problems.



Find the price elasticity of demand for a demand function.

54–58, 99 59–62

* Use a wide range of valuable study aids to help you master the material in this chapter. The Student Solutions Guide includes step-by-step solutions to all odd-numbered exercises to help you review and prepare. The student website at college.hmco.com/info/larsonapplied offers algebra help and a Graphing Technology Guide. The Graphing Technology Guide contains step-by-step commands and instructions for a wide variety of graphing calculators, including the most recent models.

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285

286

CHAPTER 3

Applications of the Derivative

Section 3.6

Review Exercises



Find the vertical and horizontal asymptotes of a function and sketch its graph.

63– 68



Find infinite limits and limits at infinity.

69–76



Use asymptotes to answer questions about real life.

77, 78

Section 3.7 ■

79–86

Analyze the graph of a function.

Section 3.8 ■

Find the differential of a function.

87–90



Use differentials to approximate changes in a function.

91–94



Use differentials to approximate changes in real-life models.

97, 98

Study Strategies ■

Solve Problems Graphically, Analytically, and Numerically When analyzing the graph of a function, use a variety of problem-solving strategies. For instance, if you were asked to analyze the graph of f 共x兲 ⫽ x3 ⫺ 4x2 ⫹ 5x ⫺ 4 you could begin graphically. That is, you could use a graphing utility to find a viewing window that appears to show the important characteristics of the graph. From the graph shown below, the function appears to have one relative minimum, one relative maximum, and one point of inflection. 1 −1

3

Relative maximum

Point of inflection

Relative minimum

−5

Next, you could use calculus to analyze the graph. Because the derivative of f is f⬘共x兲 ⫽ 3x2 ⫺ 8x ⫹ 5 ⫽ 共3x ⫺ 5兲共x ⫺ 1兲 the critical numbers of f are x ⫽ 53 and x ⫽ 1. By the First-Derivative Test, you can conclude that x ⫽ 53 yields a relative minimum and x ⫽ 1 yields a relative maximum. Because f ⬙ 共x兲 ⫽ 6x ⫺ 8 you can conclude that x ⫽ 43 yields a point of inflection. Finally, you could analyze the graph numerically. For instance, you could construct a table of values and observe that f is increasing on the interval 共⫺ ⬁, 1兲, decreasing on the interval 共1, 53 兲, and increasing on the interval 共53, ⬁兲. ■

Problem-Solving Strategies If you get stuck when trying to solve an optimization problem, consider the strategies below.

1. Draw a Diagram. If feasible, draw a diagram that represents the problem. Label all known values and unknown values on the diagram. 2. Solve a Simpler Problem. Simplify the problem, or write several simple examples of the problem. For instance, if you are asked to find the dimensions that will produce a maximum area, try calculating the areas of several examples. 3. Rewrite the Problem in Your Own Words. Rewriting a problem can help you understand it better. 4. Guess and Check. Try guessing the answer, then check your guess in the statement of the original problem. By refining your guesses, you may be able to think of a general strategy for solving the problem.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Review Exercises

Review Exercises In Exercises 1– 4, find the critical numbers of the function. 1. f 共x兲 ⫽

⫺x2

⫹ 2x ⫹ 4

2. g共x兲 ⫽ 共x ⫺ 1兲2共x ⫺ 3兲 3. h共x兲 ⫽ 冪x 共x ⫺ 3兲 4. f(x兲 ⫽ 共x ⫹ 3兲

2

In Exercises 5–8, determine the open intervals on which the function is increasing or decreasing. Solve the problem analytically and graphically. 5. f 共x兲 ⫽ x2 ⫹ x ⫺ 2

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

15. h共x兲 ⫽ 2x2 ⫺ x 4 16. s共x兲 ⫽ x 4 ⫺ 8x2 ⫹ 3 17. f 共x兲 ⫽

6 x2 ⫹ 1

18. f 共x兲 ⫽

2 x2 ⫺ 1

19. h共x兲 ⫽

x2 x⫺2

20. g共x兲 ⫽ x ⫺ 6冪x,

x > 0

In Exercises 21–30, find the absolute extrema of the function on the closed interval. Then use a graphing utility to confirm your result.

6. g共x兲 ⫽ 共x ⫹ 2兲3 7. h共x兲 ⫽

x2 ⫺ 3x ⫺ 4 x⫺3

21. f 共x兲 ⫽ x2 ⫹ 5x ⫹ 6; 关⫺3, 0兴 22. f 共x兲 ⫽ x 4 ⫺ 2x3; 关0, 2兴

8. f 共x兲 ⫽ ⫺x3 ⫹ 6x2 ⫺ 2 9. Meteorology The monthly normal temperature T (in degrees Fahrenheit) for New York City can be modeled by T ⫽ 0.0380t 4 ⫺ 1.092t 3 ⫹ 9.23t 2 ⫺ 19.6t ⫹ 44 where 1 ≤ t ≤ 12 and t ⫽ 1 corresponds to January. (Source: National Climatic Data Center) (a) Find the interval(s) on which the model is increasing.

23. f 共x兲 ⫽ x3 ⫺ 12x ⫹ 1; 关⫺4, 4兴 24. f 共x兲 ⫽ x3 ⫹ 2x2 ⫺ 3x ⫹ 4; 关⫺3, 2兴 25. f 共x兲 ⫽ 4冪x ⫺ x2; 关0, 3兴 26. f 共x兲 ⫽ 2冪x ⫺ x; 关0, 9兴 27. f 共x兲 ⫽

x 冪x2 ⫹ 1

; 关0, 2兴

(b) Find the interval(s) on which the model is decreasing.

28. f 共x兲 ⫽ ⫺x 4 ⫹ x2 ⫹ 2; 关0, 2兴

(c) Interpret the results of parts (a) and (b).

29. f 共x兲 ⫽

2x ; 关⫺1, 2兴 x2 ⫹ 1

30. f 共x兲 ⫽

8 ⫹ x; x

(d) Use a graphing utility to graph the model. 10. CD Shipments The number S of manufacturer unit shipments (in millions) of CDs in the United States from 2000 through 2005 can be modeled by S ⫽ ⫺4.17083t4 ⫹ 40.3009t3 ⫺ 110.524t 2 ⫹ 19.40t ⫹ 941.6 where 0 ≤ t ≤ 5 and t ⫽ 0 corresponds to 2000. (Source: Recording Industry Association of America) (a) Find the interval(s) on which the model is increasing. (b) Find the interval(s) on which the model is decreasing. (c) Interpret the results of parts (a) and (b). (d) Use a graphing utility to graph the model. In Exercises 11–20, use the First-Derivative Test to find the relative extrema of the function. Then use a graphing utility to verify your result. 11. f 共x兲 ⫽ 4x3 ⫺ 6x2 ⫺ 2 13. g共x兲 ⫽ x2 ⫺ 16x ⫹ 12 14. h共x兲 ⫽ 4 ⫹ 10x ⫺ x2

287

1 12. f 共x兲 ⫽ 4x 4 ⫺ 8x

关1, 4兴

31. Surface Area A right circular cylinder of radius r and height h has a volume of 25 cubic inches. The total surface area of the cylinder in terms of r is given by



S ⫽ 2␲ r r ⫹



25 . ␲ r2

Use a graphing utility to graph S and S⬘ and find the value of r that yields the minimum surface area. 32. Environment When organic waste is dumped into a pond, the decomposition of the waste consumes oxygen. A model for the oxygen level O (where 1 is the normal level) of a pond as waste material oxidizes is O⫽

t2 ⫺ t ⫹ 1 , t2 ⫹ 1

0 ≤ t

where t is the time in weeks. (a) When is the oxygen level lowest? What is this level? (b) When is the oxygen level highest? What is this level? (c) Describe the oxygen level as t increases.

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288

CHAPTER 3

Applications of the Derivative

In Exercises 33–36, determine the open intervals on which the graph of the function is concave upward or concave downward. Then use a graphing utility to confirm your result. 33. f 共x兲 ⫽ 共x ⫺ 2兲3 35. g共x兲 ⫽

1 4 4 共⫺x

34. h共x兲 ⫽ x5 ⫺ 10x2

⫹ 8x ⫺ 12兲 2

36. h共x兲 ⫽ x ⫺ 6x 3

In Exercises 37– 40, find the points of inflection of the graph of the function. 1 37. f 共x兲 ⫽ 2x 4 ⫺ 4x3 1 38. f 共x兲 ⫽ 4x 4 ⫺ 2x2 ⫺ x

39. f 共x兲 ⫽ x3共x ⫺ 3兲2 40. f 共x兲 ⫽ 共x ⫺ 1兲2共x ⫺ 3兲 In Exercises 41–44, use the Second-Derivative Test to find the relative extrema of the function. 41. f 共x兲 ⫽ x5 ⫺ 5x3 42. f 共x兲 ⫽ x 共x2 ⫺ 3x ⫺ 9兲 43. f 共x兲 ⫽ 2x2共1 ⫺ x2兲 44. f 共x兲 ⫽ x ⫺ 4冪x ⫹ 1 Point of Diminishing Returns In Exercises 45 and 46, identify the point of diminishing returns for the inputoutput function. For each function, R is the revenue (in thousands of dollars) and x is the amount spent on advertising (in thousands of dollars). 1 45. R ⫽ 1500共150x2 ⫺ x3兲, 0 ≤ x ≤ 100

46. R ⫽

⫺ 23共x3

⫺ 12x2 ⫺ 6兲, 0 ≤ x ≤ 8

47. Minimum Sum Find two positive numbers whose product is 169 and whose sum is a minimum. Solve the problem analytically, and use a graphing utility to solve the problem graphically. 48. Length The wall of a building is to be braced by a beam that must pass over a five-foot fence that is parallel to the building and 4 feet from the building. Find the length of the shortest beam that can be used. 49. Newspaper Circulation The total number N of daily newspapers in circulation (in millions) in the United States from 1970 through 2005 can be modeled by N ⫽ 0.022t ⫺ 1.27t ⫹ 9.7t ⫹ 1746 3

2

where 0 ≤ t ≤ 35 and t ⫽ 0 corresponds to 1970. (Source: Editor and Publisher Company) (a) Find the absolute maximum and minimum over the time period. (b) Find the year in which the circulation was changing at the greatest rate. (c) Briefly explain your results for parts (a) and (b).

50. Minimum Cost A fence is to be built to enclose a rectangular region of 4800 square feet. The fencing material along three sides costs $3 per foot. The fencing material along the fourth side costs $4 per foot. (a) Find the most economical dimensions of the region. (b) How would the result of part (a) change if the fencing material costs for all sides increased by $1 per foot? 51. Biology The growth of a red oak tree is approximated by the model y ⫽ ⫺0.003x3 ⫹ 0.137x2 ⫹ 0.458x ⫺ 0.839, 2 ≤ x ≤ 34 where y is the height of the tree in feet and x is its age in years. Find the age of the tree when it is growing most rapidly. Then use a graphing utility to graph the function and to verify your result. (Hint: Use the viewing window 2 ≤ x ≤ 34 and ⫺10 ≤ y ≤ 60.) 52. Consumer Trends The average number of hours N (per person per year) of TV usage in the United States from 2000 through 2005 can be modeled by N ⫽ ⫺0.382t 3 ⫺ 0.97t 2 ⫹ 30.5t ⫹ 1466, where t ⫽ 0 corresponds to 2000. Suhler Stevenson)

0 ≤ t ≤ 5

(Source: Veronis

(a) Find the intervals on which dN兾dt is increasing and decreasing. (b) Find the limit of N as t → ⬁. (c) Briefly explain your results for parts (a) and (b). 53. Medicine: Poiseuille’s Law The speed of blood that is r centimeters from the center of an artery is modeled by s共r兲 ⫽ c共R2 ⫺ r2兲,

c > 0

where c is a constant, R is the radius of the artery, and s is measured in centimeters per second. Show that the speed is a maximum at the center of an artery. 54. Profit The demand and cost functions for a product are p ⫽ 36 ⫺ 4x and C ⫽ 2x2 ⫹ 6. (a) What level of production will produce a maximum profit? (b) What level of production will produce a minimum average cost per unit? 55. Revenue For groups of 20 or more, a theater determines the ticket price p according to the formula p ⫽ 15 ⫺ 0.1共n ⫺ 20兲,

20 ≤ n ≤ N

where n is the number in the group. What should the value of N be? Explain your reasoning. 56. Minimum Cost The cost of fuel to run a locomotive is proportional to the 32 power of the speed. At a speed of 25 miles per hour, the cost of fuel is $50 per hour. Other costs amount to $100 per hour. Find the speed that will minimize the cost per mile.

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289

Review Exercises 57. Inventory Cost The cost C of inventory modeled by C⫽

冢Qx冣 s ⫹ 冢2x 冣 r

x→0

depends on ordering and storage costs, where Q is the number of units sold per year, r is the cost of storing one unit for 1 year, s is the cost of placing an order, and x is the number of units in the order. Determine the order size that will minimize the cost when Q ⫽ 10,000, s ⫽ 4.5, and r ⫽ 5.76. 58. Profit The demand and cost functions for a product are given by p ⫽ 600 ⫺ 3x

71.

P ⫽ xp ⫺ C ⫺ xt where t is the excise tax per unit. Find the maximum profits for excise taxes of t ⫽ $5, t ⫽ $10, and t ⫽ $20. In Exercises 59– 62, find the intervals on which the demand is elastic, inelastic, and of unit elasticity. 59. p ⫽ 30 ⫺ 0.2x, 0 ≤ x ≤ 150 60. p ⫽ 60 ⫺ 0.04x, 62. p ⫽ 960 ⫺ x ,

2x2 3x2 ⫹ 5

x→ ⬁

3x2 ⫺ 2x ⫹ 3 x→ ⬁ x⫹1

74. lim

76.

lim

3x x2 ⫹ 1

lim

冢x ⫺x 2 ⫹ x 2x⫹ 2冣

x→⫺⬁

x→⫺⬁

77. Health For a person with sensitive skin, the maximum amount T (in hours) of exposure to the sun that can be tolerated before skin damage occurs can be modeled by T⫽

2x ⫹ 3 x⫺4

64. g共x兲 ⫽

3 ⫺2 x

(b) Describe the value of T as s increases. Sensitive Skin T

0 ≤ x ≤ 960

x

66. h共x兲 ⫽

3x 冪x2 ⫹ 2

67. f 共x兲 ⫽

4 x2 ⫹ 1

68. h共x兲 ⫽

2x2 ⫹ 3x ⫺ 5 x⫺1

In Exercises 69–76, find the limit, if it exists. 69. lim⫹ x→0



1 x⫺ 3 x



0 < s ≤ 120

where s is the Sunsor Scale reading. (Source: Sunsor, Inc.)

0 ≤ x ≤ 300

冪9x2 ⫹ 1

65. f 共x兲 ⫽

⫺0.03s ⫹ 33.6 , s

(a) Use a graphing utility to graph the model. Compare your result with the graph below.

In Exercises 63– 68, find the vertical and horizontal asymptotes of the graph. Then use a graphing utility to graph the function. 63. h共x兲 ⫽

⫺ 2x ⫹ 1 x⫹1

73. lim

x→3

Exposure time (in hours)

61. p ⫽ 300 ⫺ x ,

0 ≤ x ≤ 1500

lim

x→⫺1 ⫹



3x2 ⫹ 1 x2 ⫺ 9

and where p is the price per unit, x is the number of units, and C is the total cost. The profit for producing x units is given by

x2

1 x

72. lim⫺

75.

C ⫽ 0.3x2 ⫹ 6x ⫹ 600



70. lim⫺ 3 ⫹

6 5 4 3 2 1 20

40

60

80

100

120

s

Sunsor Scale reading

78. Average Cost and Profit The cost and revenue functions for a product are given by C ⫽ 10,000 ⫹ 48.9x and R ⫽ 68.5x. (a) Find the average cost function. (b) What is the limit of the average cost as x approaches infinity? (c) Find the average profits when x is 1 million, 2 million, and 10 million. (d) What is the limit of the average profit as x increases without bound?

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

290

CHAPTER 3

Applications of the Derivative

In Exercises 79– 86, use a graphing utility to graph the function. Use the graph to approximate any intercepts, relative extrema, points of inflection, and asymptotes. State the domain of the function. 79. f 共x兲 ⫽ 4x ⫺ x2

80. f 共x兲 ⫽ 4x3 ⫺ x 4

81. f 共x兲 ⫽ x冪16 ⫺ x2

82. f 共x兲 ⫽ x2冪9 ⫺ x2

83. f 共x兲 ⫽

x⫹1 x⫺1

85. f 共x兲 ⫽ x2 ⫹

84. f 共x兲 ⫽

2 x

x⫺1 3x2 ⫹ 1

(d) Find the years in which the revenue per share was increasing and decreasing. (e) Find the years in which the rate of change of the revenue per share was increasing and decreasing. (f) Briefly explain your results for parts (d) and (e). 96. Medicine The effectiveness E of a pain-killing drug t hours after entering the bloodstream is modeled by

86. f 共x兲 ⫽ x 4兾5

E ⫽ 22.5t ⫹ 7.5t 2 ⫺ 2.5t 3,

0 ≤ t ≤ 4.5.

(a) Use a graphing utility to graph the equation. Choose an appropriate window.

In Exercises 87– 90, find the differential dy. 87. y ⫽ x共1 ⫺ x兲

(b) Find the maximum effectiveness the pain-killing drug attains over the interval 关0, 4.5兴.

88. y ⫽ 共3x2 ⫺ 2兲3 89. y ⫽ 冪36 ⫺ x 2 2⫺x 90. y ⫽ x⫹5 In Exercises 91–94, use differentials to approximate the change in cost, revenue, or profit corresponding to an increase in sales of one unit. 91. C ⫽ 40x2 ⫹ 1225, x ⫽ 10 3 x ⫹ 500, x ⫽ 125 92. C ⫽ 1.5 冪

93. R ⫽ 6.25x ⫹ 0.4x 3兾2,

x ⫽ 225

94. P ⫽ 0.003x2 ⫹ 0.019x ⫺ 1200, x ⫽ 750 95. Revenue Per Share The revenues per share R (in dollars) for the Walt Disney Company for the years 1994 through 2005 are shown in the table. (Source: The Walt Disney Company) Year, t

4

5

6

Revenue per share, R

6.40

7.70

10.50 11.10 11.21 11.34

7

8

9

Year, t

10

Revenue per share, R

12.09 12.52 12.40 13.23 15.05 15.91

11

12

13

14

15

(a) Use a graphing utility to create a scatter plot of the data, where t is the time in years, with t ⫽ 4 corresponding to 1994. (b) Describe any trends and/or patterns in the data. (c) A model for the data is R⫽

Graph the model and the data in the same viewing window.

5.75 ⫺ 2.043t ⫹ 0.1959t 2 , 1 ⫺ 0.378t ⫹ 0.0438t 2 ⫺ 0.00117t 3 4 ≤ t ≤ 15.

97. Surface Area and Volume The diameter of a sphere is measured to be 18 inches with a possible error of 0.05 inch. Use differentials to approximate the possible error in the surface area and the volume of the sphere. 98. Demand A company finds that the demand for its product is modeled by p ⫽ 85 ⫺ 0.125x. If x changes from 7 to 8, what is the corresponding change in p? Compare the values of ⌬p and dp. 99. Economics: Revenue Consider the following cost and demand information for a monopoly (in dollars). Complete the table, and then use the information to answer the questions. (Source: Adapted from Taylor, Economics, Fifth Edition) Quantity of output

Price

1

14.00

2

12.00

3

10.00

4

8.50

5

7.00

6

5.50

Total revenue

Marginal revenue

(a) Use the regression feature of a graphing utility to find a quadratic model for the total revenue data. (b) From the total revenue model you found in part (a), use derivatives to find an equation for the marginal revenue. Now use the values for output in the table and compare the results with the values in the marginal revenue column of the table. How close was your model? (c) What quantity maximizes total revenue for the monopoly?

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

Chapter Test

Chapter Test

291

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Take this test as you would take a test in class. When you are done, check your work against the answers given in the back of the book. In Exercises 1–3, find the critical numbers of the function and the open intervals on which the function is increasing or decreasing . 1. f 共x兲 ⫽ 3x2 ⫺ 4

2. f 共x兲 ⫽ x 3 ⫺ 12x

3. f 共x兲 ⫽ 共x ⫺ 5兲4

In Exercises 4– 6, use the First-Derivative Test to find all relative extrema of the function. Then use a graphing utility to verify your result. 1 4. f 共x兲 ⫽ x3 ⫺ 9x ⫹ 4 3

5. f 共x兲 ⫽ 2x 4 ⫺ 4x2 ⫺ 5

6. f 共x兲 ⫽

5 x2 ⫹ 2

In Exercises 7–9, find the absolute extrema of the function on the closed interval. 7. f 共x兲 ⫽ x2 ⫹ 6x ⫹ 8, 关⫺4, 0兴 9. f 共x兲 ⫽

6 x ⫹ , x 2

8. f 共x兲 ⫽ 12冪x ⫺ 4x, 关0, 5兴

关1, 6兴

In Exercises 10 and 11, determine the open intervals on which the graph of the function is concave upward or concave downward. 10. f 共x兲 ⫽ x 5 ⫺ 4x 2

11. f 共x兲 ⫽

20 3x2 ⫹ 8

In Exercises 12 and 13, find the point(s) of inflection of the graph of the function. 1 13. f 共x兲 ⫽ x5 ⫺ 4x2 5

12. f 共x兲 ⫽ x 4 ⫹ 6

In Exercises 14 and 15, use the Second-Derivative Test to find all relative extrema of the function. 3 15. f 共x兲 ⫽ x5 ⫺ 9x 3 5

14. f 共x兲 ⫽ x3 ⫺ 6x2 ⫺ 24x ⫹ 12

In Exercises 16–18, find the vertical and horizontal asymptotes of the graph. Then use a graphing utility to graph the function. 16. f 共x兲 ⫽

3x ⫹ 2 x⫺5

17. f 共x兲 ⫽

2x2 ⫹3

x2

18. f 共x兲 ⫽

2x2 ⫺ 5 x⫺1

21.

6x2 ⫹ x ⫺ 5 2x2 ⫺ 5x

In Exercises 19–21, find the limit, if it exists. 19. lim

x→ ⬁

冢3x ⫹ 1冣

20. lim

x→ ⬁

3x2 ⫺ 4x ⫹ 1 x⫺7

lim

x→⫺⬁

In Exercises 22–24, find the differential dy. 22. y ⫽ 5x2 ⫺ 3

23. y ⫽

1⫺x x⫹3

24. y ⫽ 共x ⫹ 4兲3

25. The demand function for a product is modeled by p ⫽ 250 ⫺ 0.4x, 0 ≤ x ≤ 625, where p is the price at which x units of the product are demanded by the market. Find the interval of inelasticity for the function.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

AP/Wide World Photos

4

Exponential and Logarithmic Functions

4.1 4.2 4.3

4.4 4.5

4.6

Exponential Functions Natural Exponential Functions Derivatives of Exponential Functions Logarithmic Functions Derivatives of Logarithmic Functions Exponential Growth and Decay

On May 26, 2006, Java, Indonesia experienced an earthquake measuring 6.3 on the Richter scale, a logarithmic function that serves as one way to calculate an earthquake’s magnitude. (See Section 4.5, Exercise 87.)

Applications Exponential and logarithmic functions have many real-life applications. The applications listed below represent a sample of the applications in this chapter. ■ ■ ■ ■ ■

Make a Decision: Median Sales Prices, Exercise 37, page 298 Learning Theory, Exercise 88, page 325 Consumer Trends, Exercise 85, page 334 Make a Decision: Revenue, Exercise 41, page 343 Make a Decision: Modeling Data, Exercise 52, page 343

292 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 4.1

Exponential Functions

293

Section 4.1

Exponential Functions

■ Use the properties of exponents to evaluate and simplify exponential

expressions. ■ Sketch the graphs of exponential functions.

Exponential Functions You are already familiar with the behavior of algebraic functions such as f 共x兲 ⫽ x2,

g共x兲 ⫽ 冪x ⫽ x1兾2,

and h共x兲 ⫽

1 ⫽ x⫺1 x

each of which involves a variable raised to a constant power. By interchanging roles and raising a constant to a variable power, you obtain another important class of functions called exponential functions. Some simple examples are f 共x兲 ⫽ 2 x,

g共x兲 ⫽

冢101 冣

x



1 , 10 x

and

h共x兲 ⫽ 32x ⫽ 9x.

In general, you can use any positive base a ⫽ 1 as the base of an exponential function. Definition of Exponential Function

If a > 0 and a ⫽ 1, then the exponential function with base a is given by f 共x兲 ⫽ a x.

STUDY TIP In the definition above, the base a ⫽ 1 is excluded because it yields f 共x兲 ⫽ 1x ⫽ 1. This is a constant function, not an exponential function.

When working with exponential functions, the properties of exponents, shown below, are useful. Properties of Exponents

Let a and b be positive numbers. 1. a0 ⫽ 1

2. a x a y ⫽ a x⫹y

3.

ax ⫽ a x⫺y ay

4. 共a x 兲 y ⫽ a xy

5. 共ab兲 x ⫽ a x b x

6.

冢ab冣

7. a⫺x ⫽

x



ax bx

1 ax

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294

CHAPTER 4

Exponential and Logarithmic Functions

Example 1

Applying Properties of Exponents

Simplify each expression using the properties of exponents. a. 共22兲共23兲 d.

b. 共22兲共2⫺3兲

⫺2

冢13冣

c. 共3 2兲3

32 33

e.

f. 共21兾2兲共31兾2兲

SOLUTION

a. 共22兲共23兲 ⫽ 22⫹3 ⫽ 25 ⫽ 32

✓CHECKPOINT 1

b. 共22兲共2⫺3兲 ⫽ 22⫺3 ⫽ 2⫺1 ⫽

Simplify each expression using the properties of exponents.

c. 共 兲 ⫽

a. 共32兲共33兲

b. 共32兲共3⫺1兲

d.

冢13冣

c. 共23兲2

d.

e.

32 1 ⫽ 32⫺3 ⫽ 3⫺1 ⫽ 33 3

e.

22 23

冢12冣

⫺3

f. 共21兾2兲共51兾2兲



R

100%





36

Apply Properties 2 and 7.

⫽ 729

Apply Property 4.

冢 冣

1 1 ⫽ 共1兾3兲2 1兾3

2

⫽ 32 ⫽ 9

Apply Properties 6 and 7.

Apply Properties 3 and 7.

f. 共21兾2兲共31兾2兲 ⫽ 关共2兲共3兲兴1兾2 ⫽ 61兾2 ⫽ 冪6

2⫺0.6 ⬇ 0.660,

Apply Property 5.

Example 2

共1.56兲冪2 ⬇ 1.876

␲ 0.75 ⬇ 2.360,

Dating Organic Material

50% 25%

3.125% 6.25%

12.5%

22,860

28,575

11,430

17,145

0

t

5,715

Ratio of isotopes to atoms

⫺2

32共3兲

Although Example 1 demonstrates the properties of exponents with integer and rational exponents, it is important to realize that the properties hold for all real exponents. With a calculator, you can obtain approximations of a x for any base a and any real exponent x. Here are some examples.

Organic Material 1.0 × 10 −12 0.9 × 10 −12 0.8 × 10 −12 0.7 × 10 −12 0.6 × 10 −12 0.5 × 10 −12 0.4 × 10 −12 0.3 × 10 −12 0.2 × 10 −12 0.1 × 10 −12

32 3

Apply Property 2. 1 2

Time (in years)

In living organic material, the ratio of radioactive carbon isotopes to the total number of carbon atoms is about 1 to 1012. When organic material dies, its radioactive carbon isotopes begin to decay, with a half-life of about 5715 years. This means that after 5715 years, the ratio of isotopes to atoms will have decreased to one-half the original ratio, after a second 5715 years the ratio will have decreased to one-fourth of the original, and so on. Figure 4.1 shows this decreasing ratio. The formula for the ratio R of carbon isotopes to carbon atoms is 1 1 t兾5715 R⫽ 12 10 2 where t is the time in years. Find the value of R for each period of time.

冢 冣冢 冣

F I G U R E 4 .1

✓CHECKPOINT 2

a. 10,000 years

Use the formula for the ratio of carbon isotopes to carbon atoms in Example 2 to find the value of R for each period of time.

SOLUTION

a. 5,000 years

冢101 冣冢12冣 1 1 b. R ⫽ 冢 10 冣冢 2 冣 1 1 c. R ⫽ 冢 冣冢 10 2冣 a. R ⫽

10,000兾5715

12

20,000兾5715

12

b. 15,000 years c. 30,000 years

b. 20,000 years

25,000兾5715



12

c. 25,000 years

⬇ 2.973 ⫻ 10⫺13

Ratio for 10,000 years

⬇ 8.842



10⫺14

Ratio for 20,000 years

⬇ 4.821



10⫺14

Ratio for 25,000 years

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 4.1

295

Exponential Functions

Graphs of Exponential Functions The basic nature of the graph of an exponential function can be determined by the point-plotting method or by using a graphing utility.

Example 3

Graphing Exponential Functions

Sketch the graph of each exponential function. a. f 共x兲 ⫽ 2x STUDY TIP Note that a graph of the form f 共x兲 ⫽ ax, as shown in Example 3(a), is a reflection in the y-axis of the graph of the form f 共x兲 ⫽ a⫺x, as shown in Example 3(b).

⫺2

To sketch these functions by hand, you can begin by constructing a table of values, as shown below. x

⫺3

⫺2

⫺1

0

1

2

3

4

1 8

1 4

1 2

1

2

4

8

16

g共x兲 ⫽ 2⫺x

8

4

2

1

1 2

1 4

1 8

1 16

h共x兲 ⫽ 3x

1 27

1 9

1 3

1

3

9

27

81

f 共x兲 ⫽

⫺1

0

2

−3 −2 −1

3

6

6

6

5

5

5

4

4

4

3

3

2

2

f(x) = 2 x 1

(a) ■

y

y

1

1

f 共x兲

2x

y

f 共x兲 x

c. h共x兲 ⫽ 3 x

SOLUTION

Complete the table of values for f 共x兲 ⫽ 5 x. Sketch the graph of the exponential function. ⫺3

共12 兲x ⫽ 2⫺x

The graphs of the three functions are shown in Figure 4.2. Note that the graphs of f 共x兲 ⫽ 2x and h 共x兲 ⫽ 3x are increasing, whereas the graph of g共x兲 ⫽ 2⫺x is decreasing.

✓CHECKPOINT 3

x

b. g共x兲 ⫽

2

3

g(x) =

3

x

( 12) = 2−x

2 1

1 x

−3 −2 −1

1

2

3

(b)

x

−3 −2 −1

h(x) = 3 x 1

2

x

3

(c)

FIGURE 4.2

TECHNOLOGY Try graphing the functions f 共x兲 ⫽ 2x and h共x兲 ⫽ 3x in the same viewing window, as shown at the right. From the display, you can see that the graph of h is increasing more rapidly than the graph of f .*

h(x) = 3 x

f(x) = 2 x

7

4

−3 −1

*Specific calculator keystroke instructions for operations in this and other technology boxes can be found at college.hmco.com/info/larsonapplied.

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296

CHAPTER 4

Exponential and Logarithmic Functions

The forms of the graphs in Figure 4.2 are typical of the graphs of the exponential functions y ⫽ a⫺x and y ⫽ ax, where a > 1. The basic characteristics of such graphs are summarized in Figure 4.3. y

y

(−2, 8)

8

(0, 1)

f(x) = 3 −x − 1

7

y

Graph of y = a x Domain: (− ∞, ∞) Range: (0, ∞) Intercept: (0, 1) Always increasing a x → ∞ as x → ∞ a x → 0 as x → − ∞ Continuous One-to-one

Graph of y = a − x Domain: (− ∞, ∞) Range: (0, ∞) Intercept: (0, 1) Always decreasing a − x → 0 as x → ∞ a − x → ∞ as x → −∞ Continuous One-to-one

(0, 1)

6 5

x

x

4 3

FIGURE 4.3 y ⫽ a x 共a > 1 兲

(−1, 2) 2 (0, 0) −3 −2 −1

x

3

Example 4

(1, ) (2, ) − 23

− 89

SOLUTION

✓CHECKPOINT 4 Complete the table of values for f 共x兲 ⫽ 2⫺x ⫹ 1. Sketch the graph of the function. Determine the horizontal asymptote of the graph. ⫺3

⫺2

⫺1

1

2

3

0

f 共x兲 x f 共x兲

Graphing an Exponential Function

Sketch the graph of f 共x兲 ⫽ 3⫺x ⫺ 1.

FIGURE 4.4

x

Characteristics of the Exponential Functions y ⫽ a⫺ x and

Begin by creating a table of values, as shown below.

x

⫺2

⫺1

0

1

f 共x兲

32 ⫺ 1 ⫽ 8

31 ⫺ 1 ⫽ 2

30 ⫺ 1 ⫽ 0

8 3⫺1 ⫺ 1 ⫽ ⫺ 23 3⫺2 ⫺ 1 ⫽ ⫺ 9

2

From the limit lim 共3⫺x ⫺ 1兲 ⫽ lim 3⫺x ⫺ lim 1

x→ ⬁

x→ ⬁

x→ ⬁

1 ⫽ lim x ⫺ lim 1 x→ ⬁ 3 x→ ⬁ ⫽0⫺1 ⫽ ⫺1



you can see that y ⫽ ⫺1 is a horizontal asymptote of the graph. The graph is shown in Figure 4.4.

CONCEPT CHECK 1. Complete the following: If a > 0 and a ⴝ 1, then f 冇x冈 ⴝ a x is a(n) _____ function. 2. Identify the domain and range of the exponential functions (a) y ⴝ aⴚx and (b) y ⴝ a x. 冇Assume a > 1.冈 3. As x approaches ⬁, what does aⴚx approach? 冇Assume a > 1.冈 4. Explain why 1 x is not an exponential function.

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SECTION 4.1

Skills Review 4.1

Exponential Functions

297

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 1.4 and 1.6.

In Exercises 1–6, describe how the graph of g is related to the graph of f. 1. g共x兲 ⫽ f 共x ⫹ 2兲

2. g共x兲 ⫽ ⫺f 共x兲

3. g共x兲 ⫽ ⫺1 ⫹ f 共x兲

4. g共x兲 ⫽ f 共⫺x兲

5. g共x兲 ⫽ f 共x ⫺ 1兲

6. g共x兲 ⫽ f 共x兲 ⫹ 2

In Exercises 7–10, discuss the continuity of the function. 7. f 共x兲 ⫽

x2 ⫹ 2x ⫺ 1 x⫹4

8. f 共x兲 ⫽

x2 ⫺ 3x ⫹ 1 x2 ⫹ 2

9. f 共x兲 ⫽

x 2 ⫺ 3x ⫺ 4 x2 ⫺ 1

10. f 共x兲 ⫽

x 2 ⫺ 5x ⫹ 4 x2 ⫹ 1

In Exercises 11–16, solve for x. 11. 2x ⫺ 6 ⫽ 4

12. 3x ⫹ 1 ⫽ 5

13. 共x ⫹ 4兲2 ⫽ 25

14. 共x ⫺ 2兲 ⫽ 8

15. x ⫹ 4x ⫺ 5 ⫽ 0

16. 2x2 ⫺ 3x ⫹ 1 ⫽ 0

2

2

Exercises 4.1

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1 and 2, evaluate each expression. 1. (a) 5共53兲

(b) 272兾3

(c) 643兾4

(d) 811兾2

(e) 253兾2

(f) 322兾5

2. (a)

共兲

1 3 5

(b)

7. f 共x兲 ⫽ 2x⫺1 (a) f 共3兲

共兲 共兲

1 1兾3 8 5 2 8

(c) 642兾3

(d)

(e) 1003兾2

(f) 45兾2

8. f 共x兲 ⫽ 3x⫹2 (a) f 共⫺4兲

(c) 共52兲2 4. (a)

53 56

(c) 共81兾2兲共21兾2兲 53 5. (a) 252 (c) 关共251兾2兲共52兲兴1兾3 6. (a) 共43兲共42兲 (c) 共46 兲1兾2

(b) 共52兲共5⫺3兲

冢15冣

3兾2

兲共3兲共

92兾3

32兾3



(d) 共82兲共43兲 (b)

3 (d) f 共⫺ 2 兲

1 (b) f 共⫺ 2 兲

(c) f 共2兲

5 (d) f 共⫺ 2 兲

(a) g共⫺2兲

(b) g共120兲

(c) g共12兲

(d) g共5.5兲

(c) g共60兲

(d) g共12.5兲

(b) g共180兲

11. Radioactive Decay After t years, the remaining mass y (in grams) of 16 grams of a radioactive element whose half-life is 30 years is given by

⫺2

1 (d) 共323兾2兲共2 兲

(b) 共

(c) f 共⫺2兲

10. g共x兲 ⫽ 1.075x (a) g共1.2兲

(d) 5⫺3 (b)

1 (b) f 共2 兲

9. g共x兲 ⫽ 1.05x

In Exercises 3–6, use the properties of exponents to simplify the expression. 3. (a) 共52兲共53兲

In Exercises 7–10, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places.

共14 兲2共42兲

(d) 关共8⫺1兲共82兾3兲兴3

y ⫽ 16

t兾30

冢12冣

,

t ≥ 0.

How much of the initial mass remains after 90 years? 12. Radioactive Decay After t years, the remaining mass y (in grams) of 23 grams of a radioactive element whose halflife is 45 years is given by y ⫽ 23

t兾45

冢12冣

, t ≥ 0.

How much of the initial mass remains after 150 years?

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298

CHAPTER 4

Exponential and Logarithmic Functions

In Exercises 13–18, match the function with its graph. [The graphs are labeled (a)–(f).] y

(a)

y

(b) 3

x

1

2

−1

1

−2 −2

−3 y

(c)

−1

1

−1

2

x

3

3

2

2 1

−1

1

−1

2

3

x

−2

y

(e)

−2

3

2

2

1

1 1

13. f 共x兲 ⫽ 3x 15. f 共x兲 ⫽ ⫺3

2

−1

x

where t is time in years and P is the present cost. If the price of an oil change for your car is presently $24.95, estimate the price 10 years from now. 36. Inflation Rate Repeat Exercise 35 assuming that the annual rate of inflation is 10% over the next 10 years and the approximate cost C of goods or services will be given by

x 1

2

3

x

14. f 共x兲 ⫽ 3⫺x兾2 x

17. f 共x兲 ⫽ 3⫺x ⫺ 1

35. Inflation Rate Suppose that the annual rate of inflation averages 4% over the next 10 years. With this rate of inflation, the approximate cost C of goods or services during any year in that decade will be given by C共t兲 ⫽ P共1.04兲t, 0 ≤ t ≤ 10

y

(f )

3

−1

−1

33. Property Value Suppose that the value of a piece of property doubles every 15 years. If you buy the property for $64,000, its value t years after the date of purchase should be V共t兲 ⫽ 64,000共2兲t兾15. Use the model to approximate the value of the property (a) 5 years and (b) 20 years after it is purchased. 34. Depreciation After t years, the value of a car that originally cost $16,000 depreciates so that each year it is 3 worth 4 of its value for the previous year. Find a model for V共t兲, the value of the car after t years. Sketch a graph of the model and determine the value of the car 4 years after it was purchased.

y

(d)

years, with t ⫽ 6 corresponding to 1996. Use the model to estimate the sales in the years (a) 2008 and (b) 2014. (Source: Starbucks Corp.)

16. f 共x兲 ⫽ 3 x⫺2 18. f 共x兲 ⫽ 3 x ⫹ 2

C共t兲 ⫽ P共1.10兲t,

0 ≤ t ≤ 10.

37. MAKE A DECISION: MEDIAN SALES PRICES For the years 1998 through 2005, the median sales prices y (in dollars) of one-family homes in the United States are shown in the table. (Source: U.S. Census Bureau and U.S. Department of Housing and Urban Development)

In Exercises 19–30, use a graphing utility to graph the function.

Year

1998

1999

2000

2001

19. f 共x兲 ⫽ 6 x

Price

152,500

161,000

169,000

175,200

Year

2002

2003

2004

2005

Price

187,600

195,000

221,000

240,900

21. f 共x兲 ⫽ 共

1 x 5



20. f 共x兲 ⫽ 4 x ⫽ 5⫺x

22. f 共x兲 ⫽ 共

1 x 4



23. y ⫽ 2

24. y ⫽ 4 ⫹ 3

25. y ⫽ ⫺2x

26. y ⫽ ⫺5 x

⫺x 2

⫺x 2

x⫺1

⫽ 4⫺x

x

27. y ⫽ 3

28. y ⫽ 2

1 29. s共t兲 ⫽ 4共3⫺t兲

30. s共t兲 ⫽ 2⫺t ⫹ 3

31. Population Growth The population P (in millions) of the United States from 1992 through 2005 can be modeled by the exponential function P共t兲 ⫽ 252.12共1.011兲t, where t is the time in years, with t ⫽ 2 corresponding to 1992. Use the model to estimate the population in the years (a) 2008 and (b) 2012. (Source: U.S. Census Bureau) 32. Sales The sales S (in millions of dollars) for Starbucks from 1996 through 2005 can be modeled by the exponential function S共t兲 ⫽ 182.34共1.272兲t, where t is the time in

A model for this data is given by y ⫽ 90,120共1.0649兲t, where t represents the year, with t ⫽ 8 corresponding to 1998. (a) Compare the actual prices with those given by the model. Does the model fit the data? Explain your reasoning. (b) Use a graphing utility to graph the model. (c) Use the zoom and trace features of a graphing utility to predict during which year the median sales price of one-family homes will reach $300,000.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 4.2

Natural Exponential Functions

299

Section 4.2 ■ Evaluate and graph functions involving the natural exponential function.

Natural Exponential Functions

■ Solve compound interest problems. ■ Solve present value problems.

Natural Exponential Functions TECHNOLOGY Try graphing y ⫽ 共1 ⫹ x兲1兾x with a graphing utility. Then use the zoom and trace features to find values of y near x ⫽ 0. You will find that the y-values get closer and closer to the number e ⬇ 2.71828.

In Section 4.1, exponential functions were introduced using an unspecified base a. In calculus, the most convenient (or natural) choice for a base is the irrational number e, whose decimal approximation is e ⬇ 2.71828182846. Although this choice of base may seem unusual, its convenience will become apparent as the rules for differentiating exponential functions are developed in Section 4.3. In that development, you will encounter the limit used in the definition of e. Limit Definition of e

The irrational number e is defined to be the limit of 共1 ⫹ x兲1兾x as x → 0. That is, lim 共1 ⫹ x兲1兾x ⫽ e.

x→0

y

Example 1

9

Sketch the graph of f 共x兲 ⫽ e x.

8

(2, e 2 )

7

SOLUTION

the table.

6

f(x) = e x

5 4 3 2 − 1, 1e 1 1

)

2

−3

−2

−1

Begin by evaluating the function for several values of x, as shown in

x

⫺2

⫺1

0

1

2

f 共x兲

e⫺2 ⬇ 0.135

e⫺1 ⬇ 0.368

e0 ⬇ 1

e1 ⬇ 2.718

e2 ⬇ 7.389

(1, e)

)

(− 2, e )

Graphing the Natural Exponential Function

(0, 1) 1

2

3

x

The graph of f 共x兲 ⫽ e x is shown in Figure 4.5. Note that e x is positive for all values of x. Moreover, the graph has the x-axis as a horizontal asymptote to the left. That is, lim e x ⫽ 0.

x→ ⫺⬁

FIGURE 4.5

✓CHECKPOINT 1 Complete the table of values for f 共x兲 ⫽ e⫺x. Sketch the graph of the function. x f 共x兲

⫺2

⫺1

0

1

2 ■

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300

CHAPTER 4

Exponential and Logarithmic Functions

Exponential functions are often used to model the growth of a quantity or a population. When the quantity’s growth is not restricted, an exponential model is often used. When the quantity’s growth is restricted, the best model is often a logistic growth function of the form f 共t兲 ⫽

a . 1 ⫹ be⫺kt

Graphs of both types of population growth models are shown in Figure 4.6. y

y

Exponential growth model: growth is not restricted.

When a culture is grown in a dish, the size of the dish and the available food limit the culture’s growth.

Logistic growth model: growth is restricted.

t

t

FIGURE 4.6

Growth of Bacterial Culture

Culture weight (in grams)

y

Example 2

1.25

MAKE A DECISION

1.20 1.15

Modeling a Population

A bacterial culture is growing according to the logistic growth model

1.10

y=

1.05

1.25 1 + 0.25e −0.4t

y⫽

1.00

1 2 3 4 5 6 7 8 9 10

Time (in hours)

FIGURE 4.7

t

1.25 , t ≥ 0 1 ⫹ 0.25e⫺0.4t

where y is the culture weight (in grams) and t is the time (in hours). Find the weight of the culture after 0 hours, 1 hour, and 10 hours. What is the limit of the model as t increases without bound? According to the model, will the weight of the culture reach 1.5 grams? SOLUTION

✓CHECKPOINT 2

y⫽

1.25 ⫽ 1 gram 1 ⫹ 0.25e⫺0.4共0兲

Weight when t ⫽ 0

A bacterial culture is growing according to the model

y⫽

1.25 ⬇ 1.071 grams 1 ⫹ 0.25e⫺0.4共1兲

Weight when t ⫽ 1

y⫽

1.25 ⬇ 1.244 grams 1 ⫹ 0.25e⫺0.4共10兲

Weight when t ⫽ 10

y⫽

1.50 , 1 ⫹ 0.2e⫺0.5t

t ≥ 0

where y is the culture weight (in grams) and t is the time (in hours). Find the weight of the culture after 0 hours, 1 hour, and 10 hours. What is the limit of the model as t increases without bound? ■

As t approaches infinity, the limit of y is lim

t→ ⬁

1.25 1.25 1.25 ⫽ lim ⫽ ⫽ 1.25. t→ ⬁ 1 ⫹ 共0.25兾e0.4t 兲 1 ⫹ 0.25e⫺0.4t 1⫹0

So, as t increases without bound, the weight of the culture approaches 1.25 grams. According to the model, the weight of the culture will not reach 1.5 grams. The graph of the model is shown in Figure 4.7.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 4.2

Natural Exponential Functions

301

Extended Application: Compound Interest TECHNOLOGY Use a spreadsheet software program or the table feature of a graphing utility to reproduce the table at the right. (Consult the user’s manual for a spreadsheet software program for specific instructions on how to create a table.) Do you get the same results as those shown in the table?

If P dollars is deposited in an account at an annual interest rate of r (in decimal form), what is the balance after 1 year? The answer depends on the number of times the interest is compounded, according to the formula



A⫽P 1⫹

r n



n

where n is the number of compoundings per year. The balances for a deposit of $1000 at 8%, at various compounding periods, are shown in the table. Number of times compounded per year, n

Balance (in dollars), A A ⫽ 1000 共1 ⫹

兲 ⫽ $1080.00 A ⫽ 1000 共1 ⫹ 兲 ⫽ $1081.60 0.08 4 A ⫽ 1000 共1 ⫹ 4 兲 ⬇ $1082.43 0.08 12 A ⫽ 1000 共1 ⫹ 12 兲 ⬇ $1083.00 0.08 365 A ⫽ 1000 共1 ⫹ 365 兲 ⬇ $1083.28

Annually, n ⫽ 1

0.08 2 2

Semiannually, n ⫽ 2 Quarterly, n ⫽ 4 Monthly, n ⫽ 12 Daily, n ⫽ 365

D I S C O V E RY Use a spreadsheet software program or the table feature of a graphing utility to evaluate the expression



1 1⫹ n



n

You may be surprised to discover that as n increases, the balance A approaches a limit, as indicated in the following development. In this development, let x ⫽ r兾n. Then x → 0 as n → ⬁, and you have

冢 nr 冣 r ⫽ P lim 冤 冢1 ⫹ 冣 冥 n ⫽ P冤 lim 共1 ⫹ x兲 冥

A ⫽ lim P 1 ⫹

n

n→ ⬁

n兾r r

n→ ⬁

for each value of n. n 10 100 1000 10,000 100,000

0.08 1 1

共1 ⫹ 1兾n兲n

䊏 䊏 䊏 䊏 䊏

What can you conclude? Try the same thing for negative values of n.

1兾x

x→0

r

Substitute x for r兾n.

⫽ Per. This limit is the balance after 1 year of continuous compounding. So, for a deposit of $1000 at 8%, compounded continuously, the balance at the end of the year would be A ⫽ 1000e0.08 ⬇ $1083.29. Summary of Compound Interest Formulas

Let P be the amount deposited, t the number of years, A the balance, and r the annual interest rate (in decimal form).



1. Compounded n times per year: A ⫽ P 1 ⫹

r n



nt

2. Compounded continuously: A ⫽ Pe rt

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302

CHAPTER 4

Exponential and Logarithmic Functions

The average interest rates paid by banks on savings accounts have varied greatly during the past 30 years. At times, savings accounts have earned as much as 12% annual interest and at times they have earned as little as 3%. The next example shows how the annual interest rate can affect the balance of an account.

Example 3 MAKE A DECISION

Finding Account Balances

You are creating a trust fund for your newborn nephew. You deposit $12,000 in an account, with instructions that the account be turned over to your nephew on his 25th birthday. Compare the balances in the account for each situation. Which account should you choose? a. 7%, compounded continuously b. 7%, compounded quarterly c. 11%, compounded continuously d. 11%, compounded quarterly

Account Balances

Account balance (in dollars)

A

SOLUTION

(25, 187,711.58)

200,000

a. 12,000e0.07共25兲 ⬇ 69,055.23

A = 12,000e 0.11t

175,000 150,000 125,000



A = 12,000e 0.07t

b. 12,000 1 ⫹

100,000



4共25兲

⬇ 68,017.87

c. 12,000e0.11共25兲 ⬇ 187,711.58

75,000 50,000 25,000

(25, 69,055.23) 5

10

15

20

Time (in years)

FIGURE 4.8

0.07 4

7%, compounded continuously

25

t



d. 12,000 1 ⫹

0.11 4



4共25兲

7%, compounded quarterly 11%, compounded continuously

⬇ 180,869.07

11%, compounded quarterly

The growth of the account for parts (a) and (c) is shown in Figure 4.8. Notice the dramatic difference between the balances at 7% and 11%. You should choose the account described in part (c) because it earns more money than the other accounts.

✓CHECKPOINT 3 Find the balance in an account if $2000 is deposited for 10 years at an interest rate of 9%, compounded as follows. Compare the results and make a general statement about compounding. a. quarterly

b. monthly

c. daily

d. continuously



In Example 3, note that the interest earned depends on the frequency with which the interest is compounded. The annual percentage rate is called the stated rate or nominal rate. However, the nominal rate does not reflect the actual rate at which interest is earned, which means that the compounding produced an effective rate that is larger than the nominal rate. In general, the effective rate corresponding to a nominal rate of r that is compounded n times per year is



Effective rate ⫽ ref f ⫽ 1 ⫹

r n



n

⫺ 1.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 4.2

Example 4

Natural Exponential Functions

303

Finding the Effective Rate of Interest

Find the effective rate of interest corresponding to a nominal rate of 6% per year compounded (a) annually, (b) quarterly, and (c) monthly. SOLUTION n

r n

冢 冣 ⫺1 0.06 ⫽ 冢1 ⫹ ⫺1 1 冣

a. reff ⫽ 1 ⫹

Formula for effective rate of interest

1

⫽ 1.06 ⫺ 1 ⫽ 0.06

Substitute for r and n. Simplify.

So, the effective rate is 6% per year. n



r n



0.06 4

b. reff ⫽ 1 ⫹ ⫽ 1⫹



⫺1

Formula for effective rate of interest



Substitute for r and n.

4

⫺1

⫽ 共1.015兲4 ⫺ 1

Simplify.

⬇ 0.0614 So, the effective rate is about 6.14% per year. n

冢 nr 冣 ⫺ 1 0.06 ⫽ 冢1 ⫹ ⫺1 12 冣

c. reff ⫽ 1 ⫹

Formula for effective rate of interest

12

⫽ 共1.005兲12 ⫺ 1 ⬇ 0.0617

Substitute for r and n. Simplify.

So, the effective rate is about 6.17% per year.

✓CHECKPOINT 4 Find the effective rate of interest corresponding to a nominal rate of 7% per year compounded (a) semiannually and (b) daily. ■

Present Value In planning for the future, this problem often arises: “How much money P should be deposited now, at a fixed rate of interest r, in order to have a balance of A, t years from now?” The answer to this question is given by the present value of A. To find the present value of a future investment, use the formula for compound interest as shown.



A⫽P 1⫹

r n



nt

Formula for compound interest

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304

CHAPTER 4

Exponential and Logarithmic Functions

Solving for P gives a present value of P⫽

A



r 1⫹ n



or

nt

P⫽

A 共1 ⫹ i兲N

where i ⫽ r兾n is the interest rate per compounding period and N ⫽ nt is the total number of compounding periods. You will learn another way to find the present value of a future investment in Section 6.1.

Example 5

Finding Present Value

An investor is purchasing a 12-year certificate of deposit that pays an annual percentage rate of 8%, compounded monthly. How much should the person invest in order to obtain a balance of $15,000 at maturity? Here, A ⫽ 15,000, r ⫽ 0.08, n ⫽ 12, and t ⫽ 12. Using the formula for present value, you obtain

SOLUTION

15,000 0.08 12共12兲 1⫹ 12 ⬇ 5761.72.

P⫽





Substitute for A, r, n, and t.

Simplify.

So, the person should invest $5761.72 in the certificate of deposit.

✓CHECKPOINT 5 How much money should be deposited in an account paying 6% interest compounded monthly in order to have a balance of $20,000 after 3 years?



CONCEPT CHECK 1. Can the number e be written as the ratio of two integers? Explain. 2. When a quantity’s growth is not restricted, which model is more often used: an exponential model or a logistic growth model? 3. When a quantity’s growth is restricted, which model is more often used: an exponential model or a logistic growth model? 4. Write the formula for the balance A in an account after t years with principal P and an annual interest rate r compounded continuously.

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 4.2

Skills Review 4.2

305

Natural Exponential Functions

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 1.6 and 3.6.

In Exercises 1–4, discuss the continuity of the function. 1. f 共x兲 ⫽

3x 2 ⫹ 2x ⫹ 1 x2 ⫹ 1

2. f 共x兲 ⫽

x⫹1 x2 ⫺ 4

3. f 共x兲 ⫽

x 2 ⫺ 6x ⫹ 5 x2 ⫺ 3

4. g共x兲 ⫽

x2 ⫺ 9x ⫹ 20 x⫺4

In Exercises 5–12, find the limit. 5. lim

25 1 ⫹ 4x

9. lim

3 2 ⫹ 共1兾x兲

x→ ⬁

x→ ⬁

8x3 ⫹ 2 x→ ⬁ 2x3 ⫹ x

16x x→ ⬁ 3 ⫹ x2

6. lim 10. lim

x→ ⬁

7. lim

6 1 ⫹ x⫺2

y

(c)

7 1 ⫹ 5x

(c) 共 兲

⫺2

e5 e3

(d)

e0

(b)

冢ee 冣

(d)

3. (a) 共e 2兲5兾2

4

8

3

6

2

4

5 ⫺1

2

2

−1

1 e⫺3

(b) 共e 2兲共e1兾2兲

(c) 共e⫺2兲⫺3

(d)

4. (a) 共e⫺3兲2兾3

(b)

(c) 共e⫺2兲⫺4

1

e4

e⫺1兾2

4

3

3

2

2

2 ex

2

3

2

3

1

2

3

x

y

(f )

1

1

−2

5

9. f 共x兲 ⫽ e冪x

2

5

4

5. f 共x兲 ⫽ e 2x⫹1 7. f 共x兲 ⫽

y

4

5

−3 −2 −1

In Exercises 5–10, match the function with its graph. [The graphs are labeled (a)–(f).]

3

−3 −2 −1

x

y

e5 e⫺2

(b)

2

(e)

(d) 共e⫺4兲共e⫺3兾2兲

y

y

(d)

10

(b) 共e3兲4

e3 ⫺2

(a)

x→ ⬁

x 2x

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

1. (a) 共e3兲共e4兲

(c)

12. lim

x→ ⬁

In Exercises 1– 4, use the properties of exponents to simplify the expression.

冢1e 冣

x→ ⬁

11. lim 2⫺x

Exercises 4.2

2. (a)

8. lim

x

−3 −2 −1

x

6. f 共x兲 ⫽ e⫺x兾2 8. f 共x兲 ⫽ e⫺1兾x 10. f 共x兲 ⫽ ⫺e x ⫹ 1

1 −3 −2 −1

1

2

3

x

−2 −3 −4

3

In Exercises 11–14, sketch the graph of the function.

2

11. h共x兲 ⫽ e x⫺3

12. f 共x兲 ⫽ e 2x

13. g共x兲 ⫽ e1⫺x

14. j共x兲 ⫽ e⫺x⫹2

1 −3 −2 − 1

1

2

3

x

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306

CHAPTER 4

Exponential and Logarithmic Functions

In Exercises 15–18, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. 15. N共t兲 ⫽ 500e⫺0.2t 17. g共x兲 ⫽

16. A共t兲 ⫽ 500e0.15t

2 2 1 ⫹ ex

18. g共x兲 ⫽

10 1 ⫹ e⫺x

In Exercises 19–22, use a graphing utility to graph the function. Determine whether the function has any horizontal asymptotes and discuss the continuity of the function. 19. f 共x兲 ⫽

e x ⫹ e⫺x 2

20. f 共x兲 ⫽

e x ⫺ e⫺x 2

21. f 共x兲 ⫽

2 1 ⫹ e1兾x

22. f 共x兲 ⫽

2 1 ⫹ 2e⫺0.2x

23. Use a graphing utility to graph f 共x兲 ⫽ ex and the given function in the same viewing window. How are the two graphs related? 1 (b) h共x兲 ⫽ ⫺ e x 2

(a) g共x兲 ⫽ e x⫺2 (c) q共x兲 ⫽ e x ⫹ 3

24. Use a graphing utility to graph the function. Describe the shape of the graph for very large and very small values of x. (a) f 共x兲 ⫽

8 1 ⫹ e⫺0.5x

(b) g共x兲 ⫽

8 1 ⫹ e⫺0.5兾x

Compound Interest In Exercises 25–28, use a spreadsheet to complete the table to determine the balance A for P dollars invested at rate r for t years, compounded n times per year. n

1

2

4

12

365

Continuous compounding

A

30. r ⫽ 3%, compounded continuously 31. r ⫽ 5%, compounded monthly 32. r ⫽ 6%, compounded daily 33. Trust Fund On the day of a child’s birth, a deposit of $20,000 is made in a trust fund that pays 8% interest, compounded continuously. Determine the balance in this account on the child’s 21st birthday. 34. Trust Fund A deposit of $10,000 is made in a trust fund that pays 7% interest, compounded continuously. It is specified that the balance will be given to the college from which the donor graduated after the money has earned interest for 50 years. How much will the college receive? 35. Effective Rate Find the effective rate of interest corresponding to a nominal rate of 9% per year compounded (a) annually, (b) semiannually, (c) quarterly, and (d) monthly. 36. Effective Rate Find the effective rate of interest corresponding to a nominal rate of 7.5% per year compounded (a) annually, (b) semiannually, (c) quarterly, and (d) monthly. 37. Present Value How much should be deposited in an account paying 7.2% interest compounded monthly in order to have a balance of $15,503.77 three years from now? 38. Present Value How much should be deposited in an account paying 7.8% interest compounded monthly in order to have a balance of $21,154.03 four years from now? 39. Future Value Find the future value of an $8000 investment if the interest rate is 4.5% compounded monthly for 2 years. 40. Future Value Find the future value of a $6500 investment if the interest rate is 6.25% compounded monthly for 3 years. 41. Demand The demand function for a product is modeled by



p ⫽ 5000 1 ⫺

25. P ⫽ $1000, r ⫽ 3%, t ⫽ 10 years 26. P ⫽ $2500, r ⫽ 2.5%, t ⫽ 20 years 27. P ⫽ $1000, r ⫽ 4%, t ⫽ 20 years 28. P ⫽ $2500, r ⫽ 5%, t ⫽ 40 years Compound Interest In Exercises 29–32, use a spreadsheet to complete the table to determine the amount of money P that should be invested at rate r to produce a final balance of $100,000 in t years. t

29. r ⫽ 4%, compounded continuously

1

10

20

30

40

50



4 . 4 ⫹ e⫺0.002x

Find the price of the product if the quantity demanded is (a) x ⫽ 100 units and (b) x ⫽ 500 units. What is the limit of the price as x increases without bound? 42. Demand The demand function for a product is modeled by



p ⫽ 10,000 1 ⫺



3 . 3 ⫹ e⫺0.001x

Find the price of the product if the quantity demanded is (a) x ⫽ 1000 units and (b) x ⫽ 1500 units. What is the limit of the price as x increases without bound?

P

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SECTION 4.2 43. Probability The average time between incoming calls at a switchboard is 3 minutes. If a call has just come in, the probability that the next call will come within the next t minutes is P共t 兲 ⫽ 1 ⫺ e⫺t兾3. Find the probability of each situation. (a) A call comes in within 12 minute. (b) A call comes in within 2 minutes. (c) A call comes in within 5 minutes. 44. Consumer Awareness An automobile gets 28 miles per gallon at speeds of up to and including 50 miles per hour. At speeds greater than 50 miles per hour, the number of miles per gallon drops at the rate of 12% for each 10 miles per hour. If s is the speed (in miles per hour) and y is the number of miles per gallon, then y ⫽ 28e0.6⫺0.012s, s > 50. Use this information and a spreadsheet to complete the table. What can you conclude? Speed (s)

50

55

60

65

70

Miles per gallon (y) 45. MAKE A DECISION: SALES The sales S (in millions of dollars) for Avon Products from 1998 through 2005 are shown in the table. (Source: Avon Products Inc.) t

8

9

10

11

S

5212.7

5289.1

5673.7

5952.0

t

12

13

14

15

S

6170.6

6804.6

7656.2

8065.2

A model for this data is given by S ⫽ 2962.6e0.0653t, where t represents the year, with t ⫽ 8 corresponding to 1998. (a) How well does the model fit the data?

Natural Exponential Functions

307

47. Biology The population y of a bacterial culture is modeled by the logistic growth function y ⫽ 925兾共1 ⫹ e⫺0.3t 兲, where t is the time in days. (a) Use a graphing utility to graph the model. (b) Does the population have a limit as t increases without bound? Explain your answer. (c) How would the limit change if the model were y ⫽ 1000兾共1 ⫹ e⫺0.3t 兲 ? Explain your answer. Draw some conclusions about this type of model. 48. Biology: Cell Division Suppose that you have a single imaginary bacterium able to divide to form two new cells every 30 seconds. Make a table of values for the number of individuals in the population over 30-second intervals up to 5 minutes. Graph the points and use a graphing utility to fit an exponential model to the data. (Source: Adapted from Levine/Miller, Biology: Discovering Life, Second Edition) 49. Learning Theory In a learning theory project, the proportion P of correct responses after n trials can be modeled by P⫽

0.83 . 1 ⫹ e⫺0.2n

(a) Use a graphing utility to estimate the proportion of correct responses after 10 trials. Verify your result analytically. (b) Use a graphing utility to estimate the number of trials required to have a proportion of correct responses of 0.75. (c) Does the proportion of correct responses have a limit as n increases without bound? Explain your answer. 50. Learning Theory In a typing class, the average number N of words per minute typed after t weeks of lessons can be modeled by N⫽

95 . 1 ⫹ 8.5e⫺0.12t

(b) Find a linear model for the data. How well does the linear model fit the data? Which model, exponential or linear, is a better fit?

(a) Use a graphing utility to estimate the average number of words per minute typed after 10 weeks. Verify your result analytically.

(c) Use the exponential growth model and the linear model from part (b) to predict when the sales will exceed 10 billion dollars.

(b) Use a graphing utility to estimate the number of weeks required to achieve an average of 70 words per minute.

46. Population The population P (in thousands) of Las Vegas, Nevada from 1960 through 2005 can be modeled by P ⫽ 68.4e0.0467t, where t is the time in years, with t ⫽ 0 corresponding to 1960. (Source: U.S. Census Bureau) (a) Find the populations in 1960, 1970, 1980, 1990, 2000, and 2005. (b) Explain why the data do not fit a linear model. (c) Use the model to estimate when the population will exceed 900,000.

(c) Does the number of words per minute have a limit as t increases without bound? Explain your answer. 51. MAKE A DECISION: CERTIFICATE OF DEPOSIT You want to invest $5000 in a certificate of deposit for 12 months. You are given the options below. Which would you choose? Explain. (a) r ⫽ 5.25%, quarterly compounding (b) r ⫽ 5%, monthly compounding (c) r ⫽ 4.75%, continuous compounding

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308

CHAPTER 4

Exponential and Logarithmic Functions

Section 4.3

Derivatives of Exponential Functions

■ Find the derivatives of natural exponential functions. ■ Use calculus to analyze the graphs of functions that involve the natural

exponential function. ■ Explore the normal probability density function.

Derivatives of Exponential Functions D I S C O V E RY Use a spreadsheet software program to compare the expressions e⌬x and 1 ⫹ ⌬x for values of ⌬x near 0. ⌬x

e⌬ x

1 ⫹ ⌬x

0.1

In Section 4.2, it was stated that the most convenient base for exponential functions is the irrational number e. The convenience of this base stems primarily from the fact that the function f 共x兲 ⫽ e x is its own derivative. You will see that this is not true of other exponential functions of the form y ⫽ a x where a ⫽ e. To verify that f 共x兲 ⫽ e x is its own derivative, notice that the limit lim 共1 ⫹ ⌬x兲1兾⌬x ⫽ e

⌬x→0

implies that for small values of ⌬x, e ⬇ 共1 ⫹ ⌬x兲1兾⌬x, or e⌬x ⬇ 1 ⫹ ⌬x. This approximation is used in the following derivation. f 共x ⫹ ⌬x兲 ⫺ f 共x兲 ⌬x x⫹⌬x ⫺ e x e ⫽ lim ⌬x→0 ⌬x x共e⌬x ⫺ 1兲 e ⫽ lim ⌬x→0 ⌬x x 关共1 ⫹ ⌬x兲 ⫺ 1兴 e ⫽ lim ⌬ x→0 ⌬x x共⌬x兲 e ⫽ lim ⌬x→0 ⌬x x ⫽ lim e

0.01

f⬘共x兲 ⫽ lim

⌬x→0

0.001 What can you conclude? Explain how this result is used in the development of the derivative of f 共x兲 ⫽ e x.

⌬x→0

⫽ ex

Definition of derivative Use f 共x兲 ⫽ e x. Factor numerator. Substitute 1 ⫹ ⌬x for e⌬x. Divide out like factor. Simplify. Evaluate limit.

If u is a function of x, you can apply the Chain Rule to obtain the derivative of e u with respect to x. Both formulas are summarized below. Derivative of the Natural Exponential Function

Let u be a differentiable function of x. 1.

d x 关e 兴 ⫽ e x dx

2.

d u du 关e 兴 ⫽ eu dx dx

TECHNOLOGY Let f 共x兲 ⫽ e x. Use a graphing utility to evaluate f 共x兲 and the numerical derivative of f 共x兲 at each x-value. Explain the results. a. x ⫽ ⫺2

b. x ⫽ 0

c. x ⫽ 2

Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

SECTION 4.3

Example 1

Derivatives of Exponential Functions

309

Interpreting a Derivativeically

Find the slopes of the tangent lines to f 共x兲 ⫽ e x

At the point (1, e) the slope is e ≈ 2.72.

at the points 共0, 1兲 and 共1, e兲. What conclusion can you make?

y

SOLUTION 4

Derivative

it follows that the slope of the tangent line to the graph of f is f⬘共0兲 ⫽ e0 ⫽ 1

2

−2

Because the derivative of f is

f⬘共x兲 ⫽ e x

3

f(x) = e x

Original function

at the point 共0, 1兲 and

At the point (0, 1) the slope is 1.

1

−1

1

2

FIGURE 4.9

Slope at point 共0, 1兲

f⬘共1兲 ⫽ e 1 ⫽ e x

Slope at point 共1, e兲

at the point 共1, e兲, as shown in Figure 4.9. From this pattern, you can see that the slope of the tangent line to the graph of f 共x兲 ⫽ e x at any point 共x, e x兲 is equal to the y-coordinate of the point.

✓CHECKPOINT 1 Find the equations of the tangent lines to f 共x兲 ⫽ e x at the points 共0, 1兲 and 共1, e兲. ■ STUDY TIP In Example 2, notice that when you differentiate an exponential function, the exponent does not change. For instance, the derivative of y ⫽ e3x is y⬘ ⫽ 3e3x. In both the function and its derivative, the exponent is 3x.

Example 2

Differentiating Exponential Functions

Differentiate each function. a. f 共x兲 ⫽ e2x c. f 共x兲 ⫽ 6e x

b. f 共x兲 ⫽ e⫺3x

2

d. f 共x兲 ⫽ e⫺x

3

SOLUTION

a. Let u ⫽ 2x. Then du兾dx ⫽ 2, and you can apply the Chain Rule. f⬘共x兲 ⫽ eu

du ⫽ e 2x共2兲 ⫽ 2e 2x dx

b. Let u ⫽ ⫺3x 2. Then du兾dx ⫽ ⫺6x, and you can apply the Chain Rule.

✓CHECKPOINT 2 Differentiate each function.

c. f 共x兲 ⫽

3

c. Let u ⫽ x 3. Then du兾dx ⫽ 3x 2, and you can apply the Chain Rule. du 3 3 ⫽ 6e x 共3x 2兲 ⫽ 18x 2e x dx

d. Let u ⫽ ⫺x. Then du兾dx ⫽ ⫺1, and you can apply the Chain Rule.

2 4e x

d. f 共x兲 ⫽ e⫺2x

du 2 2 ⫽ e⫺3x 共⫺6x兲 ⫽ ⫺6xe⫺3x dx

f⬘共x兲 ⫽ 6eu

a. f 共x兲 ⫽ e3x b. f 共x兲 ⫽ e⫺2x

f⬘共x兲 ⫽ eu



f⬘共x兲 ⫽ eu

du ⫽ e⫺x共⫺1兲 ⫽ ⫺e⫺x dx

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310

CHAPTER 4

Exponential and Logarithmic Functions

The differentiation rules that you studied in Chapter 2 can be used with exponential functions, as shown in Example 3.

Example 3

Differentiating Exponential Functions

Differentiate each function. a. f 共x兲 ⫽ xe x c. f 共x兲 ⫽

b. f 共x兲 ⫽

ex x

e x ⫺ e ⫺x 2

d. f 共x兲 ⫽ xe x ⫺ e x

SOLUTION

a. f 共x兲 ⫽ xe x f⬘共x兲 ⫽ xe x ⫹ e x共1兲 ⫽ xe x ⫹ e x

Write original function. Product Rule Simplify.

e⫺x

e ⫺ 2 1 x ⫽ 2共e ⫺ e⫺x兲

b. f 共x兲 ⫽ f⬘共x兲 ⫽ c. f 共x兲 ⫽

x

1 x 2 共e



e⫺x

Write original function. Rewrite.



Constant Multiple Rule

ex x

Write original function.

xe x ⫺ e x共1兲 x2 x共x ⫺ 1兲 e ⫽ x2

f⬘共x兲 ⫽

Quotient Rule Simplify.

d. f 共x兲 ⫽ xe x ⫺ e x f⬘共x兲 ⫽ 关xe x ⫹ e x共1兲兴 ⫺ e x ⫽ xe x ⫹ e x ⫺ e x ⫽ xe x

Write original function. Product and Difference Rules

Simplify.

✓CHECKPOINT 3 Differentiate each function. a. f 共x兲 ⫽ x2e x c. f 共x兲 ⫽

ex x2

b. f 共x兲 ⫽

e x ⫹ e⫺x 2

d. f 共x兲 ⫽ x2e x ⫺ e x



TECHNOLOGY If you have access to a symbolic differentiation utility, try using it to find the derivatives of the functions in Example 3.

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SECTION 4.3

Derivatives of Exponential Functions

311

Applications In Chapter 3, you learned how to use derivatives to analyze the graphs of functions. The next example applies those techniques to a function composed of exponential functions. In the example, notice that e a ⫽ e b implies that a ⫽ b.

Example 4

Analyzing a Catenary

When a telephone wire is hung between two poles, the wire forms a U-shaped curve called a catenary. For instance, the function y ⫽ 30共e x兾60 ⫹ e⫺x兾60兲,

⫺30 ≤ x ≤ 30

models the shape of a telephone wire strung between two poles that are 60 feet apart (x and y are measured in feet). Show that the lowest point on the wire is midway between the two poles. How much does the wire sag between the two poles? © Don Hammond/Design Pics/Corbis

Utility wires strung between poles have the shape of a catenary.

SOLUTION

The derivative of the function is

1 y⬘ ⫽ 30关e x兾60共60 兲 ⫹ e⫺ x兾60共⫺ 601 兲兴

⫽ 12共e x兾60 ⫺ e⫺x兾60兲. To find the critical numbers, set the derivative equal to zero. 1 x兾60 2 共e

⫺ e⫺x兾60兲 ⫽ 0 e x兾60 ⫺ e⫺x兾60 ⫽ 0 e x兾60 ⫽ e⫺x兾60 x x ⫽⫺ 60 60 x ⫽ ⫺x 2x ⫽ 0 x⫽0

y

80

20

FIGURE 4.10

Multiply each side by 2. Add e⫺x兾60 to each side. If ea ⫽ eb, then a ⫽ b. Multiply each side by 60. Add x to each side. Divide each side by 2.

Using the First-Derivative Test, you can determine that the critical number x ⫽ 0 yields a relative minimum of the function. From the graph in Figure 4.10, you can see that this relative minimum is actually a minimum on the interval 关⫺30, 30兴. To find how much the wire sags between the two poles, you can compare its height at each pole with its height at the midpoint.

40

− 30

Set derivative equal to 0.

30

x

y ⫽ 30共e⫺30兾60 ⫹ e⫺共⫺30兲兾60兲 ⬇ 67.7 feet y ⫽ 30共e0兾60 ⫹ e⫺共0兲兾60兲 ⫽ 60 feet y ⫽ 30共e30兾60 ⫹ e⫺共30兲兾60兲 ⬇ 67.7 feet

Height at left pole Height at midpoint Height at right pole

From this, you can see that the wire sags about 7.7 feet.

✓CHECKPOINT 4 Use a graphing utility to graph the function in Example 4. Verify the minimum value. Use the information in the example to choose an appropriate viewing window. ■

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312

CHAPTER 4

Exponential and Logarithmic Functions

Example 5

Finding a Maximum Revenue

The demand function for a product is modeled by p ⫽ 56e⫺0.000012x

Demand function

where p is the price per unit (in dollars) and x is the number of units. What price will yield a maximum revenue? SOLUTION

The revenue function is

R ⫽ xp ⫽ 56xe⫺0.000012x.

Revenue function

To find the maximum revenue analytically, you would set the marginal revenue, dR兾dx, equal to zero and solve for x. In this problem, it is easier to use a graphical approach. After experimenting to find a reasonable viewing window, you can obtain a graph of R that is similar to that shown in Figure 4.11. Using the zoom and trace features, you can conclude that the maximum revenue occurs when x is about 83,300 units. To find the price that corresponds to this production level, substitute x ⬇ 83,300 into the demand function. p ⬇ 56e⫺0.000012共83,300兲 ⬇ $20.61. So, a price of about $20.61 will yield a maximum revenue. 2,000,000

Maximum revenue

0

500,000

0

F I G U R E 4 . 1 1 Use the zoom and trace features to approximate the x-value that corresponds to the maximum revenue.

✓CHECKPOINT 5 The demand function for a product is modeled by p ⫽ 50e⫺0.0000125x where p is the price per unit in dollars and x is the number of units. What price will yield a maximum revenue? ■ STUDY TIP Try solving the problem in Example 5 analytically. When you do this, you obtain dR ⫽ 56xe⫺0.000012x共⫺0.000012兲 ⫹ e⫺0.000012x共56兲 ⫽ 0. dx Explain how you would solve this equation. What is the solution?

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SECTION 4.3

Derivatives of Exponential Functions

313

The Normal Probability Density Function If you take a course in statistics or quantitative business analysis, you will spend quite a bit of time studying the characteristics and use of the normal probability density function given by f 共x兲 ⫽ Two points of inflection 0.5

y

1 e −x 2/2 2π

f(x) =

0.3 0.2 0.1 −2

−1

1 2 2 e⫺共x⫺ ␮兲 兾2␴ ␴冪2␲

where ␴ is the lowercase Greek letter sigma, and ␮ is the lowercase Greek letter mu. In this formula, ␴ represents the standard deviation of the probability distribution, and ␮ represents the mean of the probability distribution.

Example 6 1

2

Exploring a Probability Density Function

x

Show that the graph of the normal probability density function

FIGURE 4.12

The graph of the normal probability density function is bell-shaped.

f 共x兲 ⫽

1

e⫺x 兾2 2

冪2␲

Original function

has points of inflection at x ⫽ ± 1. SOLUTION

✓CHECKPOINT 6 Graph the normal probability density function f 共x兲 ⫽

1 2 e⫺x 兾32 4冪2␲

and approximate the points of inflection. ■

Begin by finding the second derivative of the function.

1 2 共⫺x兲e⫺x 兾2 冪2␲ 1 2 2 f ⬙ 共x兲 ⫽ 关共⫺x兲共⫺x兲e⫺x 兾2 ⫹ 共⫺1兲e⫺x 兾2兴 冪2␲ 1 2 ⫽ 共e⫺x 兾2兲共x2 ⫺ 1兲 冪2␲ f⬘共x兲 ⫽

First derivative Second derivative Simplify.

By setting the second derivative equal to 0, you can determine that x ⫽ ± 1. By testing the concavity of the graph, you can then conclude that these x-values yield points of inflection, as shown in Figure 4.12.

CONCEPT CHECK 1. What is the derivative of f 冇x冈 ⴝ e x? 2. What is the derivative of f 冇x冈 ⴝ eu? 冇Assume that u is a differentiable function of x.冈 3. If ea ⴝ eb, then a is equal to what? 4. In the normal probability density function given by f 冇x冈 ⴝ

1 2 2 eⴚ冇xⴚ ␮冈 /2␴ ␴冪2␲

identify what is represented by (a) ␴ and (b) ␮.

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314

CHAPTER 4

Skills Review 4.3

Exponential and Logarithmic Functions The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 0.4, 2.4, and 3.2.

In Exercises 1–4, factor the expression. 1 1. x2ex ⫺ 2e x

2. 共xe⫺x兲⫺1 ⫹ e x

3. xe x ⫺ e 2x

4. e x ⫺ xe⫺x

In Exercises 5–8, find the derivative of the function. 3 x 5. f 共x兲 ⫽ 2 6. g共x兲 ⫽ 3x 2 ⫺ 7x 6 7. f 共x兲 ⫽ 共4x ⫺ 3兲共x2 ⫹ 9兲

8. f 共t兲 ⫽

t⫺2 冪t

In Exercises 9 and 10, find the relative extrema of the function. 1 9. f 共x兲 ⫽ 8 x3 ⫺ 2 x

10. f 共x兲 ⫽ x 4 ⫺ 2x 2 ⫹ 5

Exercises 4.3

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–4, find the slope of the tangent line to the exponential function at the point 冇0, 1冈. 1. y ⫽ e 3x

2. y ⫽ e 2x

y

In Exercises 17–22, determine an equation of the tangent line to the function at the given point. 17. y ⫽ e⫺2x⫹x , 2

y

19. y ⫽ x 2 e⫺x, (0, 1)

(0, 1) x

1

3. y ⫽ e⫺x

冢2, e4 冣

21. y ⫽ 共e 2x ⫹ 1兲3, x

1

4. y ⫽ e⫺2x y

共2, 1兲

18. g共x兲 ⫽ e x , 3

x , e2x

冢⫺1, 1e 冣

冢1, e1 冣

2

20. y ⫽

共0, 8兲

22. y ⫽ 共e4x ⫺ 2兲2,

2

共0, 1兲

In Exercises 23–26, find dy/dx implicitly. 23. xey ⫺ 10x ⫹ 3y ⫽ 0

24. x2y ⫺ ey ⫺ 4 ⫽ 0

25. x 2e⫺x ⫹ 2y 2 ⫺ xy ⫽ 0

26. e xy ⫹ x 2 ⫺ y 2 ⫽ 10

y

In Exercises 27–30, find the second derivative. (0, 1)

1

−1

1

(0, 1)

x

1

x

In Exercises 5–16, find the derivative of the function. 5. y ⫽ e 7. y ⫽

6. y ⫽ e

5x

2 e⫺x

8. f 共x兲 ⫽ e1兾x

⫺1兾x 2

9. f 共x兲 ⫽ e

11. f 共x兲 ⫽ 共x 2 ⫹ 1兲e 4x 13. f 共x兲 ⫽

ex



1⫺x

2 ⫹ e⫺x 兲 3

15. y ⫽ xe x ⫺ 4e⫺x

10. g共x兲 ⫽ e冪x 12. y ⫽ 4x3e⫺x 14. f 共x兲 ⫽

共e x ⫹ e⫺x兲4 2

27. f 共x兲 ⫽ 2e 3x ⫹ 3e⫺2x

28. f 共x兲 ⫽ 共1 ⫹ 2x兲e 4x

29. f 共x兲 ⫽ 5e⫺x ⫺ 2e⫺5x

30. f 共x兲 ⫽ 共3 ⫹ 2x兲e⫺3x

In Exercises 31–34, graph and analyze the function. Include extrema, points of inflection, and asymptotes in your analysis. 31. f 共x兲 ⫽

1 2 ⫺ e⫺x

33. f 共x兲 ⫽ x 2e⫺x

32. f 共x兲 ⫽

e x ⫺ e⫺x 2

34. f 共x兲 ⫽ xe⫺x

In Exercises 35 and 36, use a graphing utility to graph the function. Determine any asymptotes of the graph. 35. f 共x兲 ⫽

8 1 ⫹ e⫺0.5x

36. g共x兲 ⫽

8 1 ⫹ e⫺0.5兾x

16. y ⫽ x 2 e x ⫺ 2xe x ⫹ 2e x

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SECTION 4.3 In Exercises 37– 40, solve the equation for x. 37.

e⫺3x

⫽e

39.

e冪x

e3



38.

ex

40.

e⫺1兾x

⫽1 ⫽ e1兾2

Derivatives of Exponential Functions

315

48. Cell Sites A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. From 1985 through 2006, the numbers y of cell sites can be modeled by 222,827 1 ⫹ 2677e⫺0.377t

Depreciation In Exercises 41 and 42, the value V (in dollars) of an item is a function of the time t (in years).

y⫽

(a) Sketch the function over the interval [0, 10]. Use a graphing utility to verify your graph.

where t represents the year, with t ⫽ 5 corresponding to 1985. (Source: Cellular Telecommunications & Internet Association)

(b) Find the rate of change of V when t ⴝ 1. (c) Find the rate of change of V when t ⴝ 5. (d) Use the values 冇0, V 冇0冈冈 and 冇10, V冇10冈冈 to find the linear depreciation model for the item. (e) Compare the exponential function and the model from part (d). What are the advantages of each? 41. V ⫽ 15,000e⫺0.6286t

42. V ⫽ 500,000e⫺0.2231t

43. Learning Theory The average typing speed N (in words per minute) after t weeks of lessons is modeled by N⫽

95 . 1 ⫹ 8.5e⫺0.12t

Find the rates at which the typing speed is changing when (a) t ⫽ 5 weeks, (b) t ⫽ 10 weeks, and (c) t ⫽ 30 weeks.

(a) Use a graphing utility to graph the model. (b) Use the graph to estimate when the rate of change in the number of cell cites began to decrease. (c) Confirm the result of part (b) analytically. 49. Probability A survey of high school seniors from a certain school district who took the SAT has determined that the mean score on the mathematics portion was 650 with a standard deviation of 12.5. (a) Assuming the data can be modeled by a normal probability density function, find a model for these data. (b) Use a graphing utility to graph the model. Be sure to choose an appropriate viewing window. (c) Find the derivative of the model. (d) Show that f⬘ > 0 for x < ␮ and f⬘ < 0 for x > ␮.

44. Compound Interest The balance A (in dollars) in a savings account is given by A ⫽ 5000e0.08t, where t is measured in years. Find the rates at which the balance is changing when (a) t ⫽ 1 year, (b) t ⫽ 10 years, and (c) t ⫽ 50 years.

50. Probability A survey of a college freshman class has determined that the mean height of females in the class is 64 inches with a standard deviation of 3.2 inches.

45. Ebbinghaus Model The Ebbinghaus Model for human memory is p ⫽ 共100 ⫺ a兲e⫺bt ⫹ a, where p is the percent retained after t weeks. (The constants a and b vary from one person to another.) If a ⫽ 20 and b ⫽ 0.5, at what rate is information being retained after 1 week? After 3 weeks?

(b) Use a graphing utility to graph the model. Be sure to choose an appropriate viewing window.

46. Agriculture The yield V (in pounds per acre) for an orchard at age t (in years) is modeled by V ⫽ 7955.6e⫺0.0458兾t. At what rate is the yield changing when (a) t ⫽ 5 years, (b) t ⫽ 10 years, and (c) t ⫽ 25 years? 47. Employment From 1996 through 2005, the numbers y (in millions) of employed people in the United States can be modeled by y ⫽ 98.020 ⫹ 6.2472t ⫺ 0.24964t 2 ⫹ 0.000002e t where t represents the year, with t ⫽ 6 corresponding to 1996. (Source: U.S. Bureau of Labor Statistics) (a) Use a graphing utility to graph the model. (b) Use the graph to estimate the rates of change in the number of employed people in 1996, 2000, and 2005. (c) Confirm the results of part (b) analytically.

(a) Assuming the data can be modeled by a normal probability density function, find a model for these data.

(c) Find the derivative of the model. (d) Show that f⬘ > 0 for x < ␮ and f⬘ < 0 for x > ␮. 51. Use a graphing utility to graph the normal probability density function with ␮ ⫽ 0 and ␴ ⫽ 2, 3, and 4 in the same viewing window. What effect does the standard deviation ␴ have on the function? Explain your reasoning. 52. Use a graphing utility to graph the normal probability density function with ␴ ⫽ 1 and ␮ ⫽ ⫺2, 1, and 3 in the same viewing window. What effect does the mean ␮ have on the function? Explain your reasoning. 53. Use Example 6 as a model to show that the graph of the normal probability density function with ␮ ⫽ 0 1 2 2 e⫺x 兾2␴ f 共x兲 ⫽ 冪 ␴ 2␲ has points of inflection at x ⫽ ± ␴. What is the maximum value of the function? Use a graphing utility to verify your answer by graphing the function for several values of ␴.

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316

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Exponential and Logarithmic Functions

Mid-Chapter Quiz

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Take this quiz as you would take a quiz in class. When you are done, check your work against the answers given in the back of the book. In Exercises 1–4, evaluate each expression. 1. 4共42兲

2.

冢23冣

3. 811兾3

4.

冢49冣

3

2

In Exercises 5–12, use properties of exponents to simplify the expression. 1 ⫺3 5. 43共42兲 6. 6

冢冣

7.

38 35

8. 共51兾2兲共31兾2兲

9. 共e2兲共e5兲 11.

10. 共e2兾3兲共e3兲

e2 e⫺4

12. 共e⫺1兲⫺3

In Exercises 13–18, use a graphing utility to graph the function. 13. f 共x兲 ⫽ 3x ⫺ 2

14. f 共x兲 ⫽ 5⫺x ⫹ 2

15. f 共x兲 ⫽ 6x⫺3

16. f 共x兲 ⫽ ex⫹2

17. f 共x兲 ⫽ 250e0.15x

18. f 共x兲 ⫽

5 1 ⫹ ex

19. Suppose that the annual rate of inflation averages 4.5% over the next 10 years. With this rate of inflation, the approximate cost C of goods or services during any year in that decade will be given by C共t兲 ⫽ P共1.045兲t,

0 ≤ t ≤ 10

where t is time in years and P is the present cost. If the price of a baseball game ticket is presently $14.95, estimate the price 10 years from now. 20. For P ⫽ $3000, r ⫽ 3.5%, and t ⫽ 5 years, find the balance in an account if interest is compounded (a) monthly and (b) continuously. In Exercises 21–24, find the derivative of the function. 21. y ⫽ e5x

22. y ⫽ ex⫺4

23. y ⫽ 5e x⫹2

24. y ⫽ 3e x ⫺ xe x

25. Determine an equation of the tangent line to y ⫽ e⫺2x at the point 共0, 1兲. 26. Graph and analyze the function f 共x兲 ⫽ 0.5x2e⫺0.5x. Include extrema, points of inflection, and asymptotes in your analysis.

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S E C T I O N 4 . 4 Logarithmic Functions

317

Section 4.4 ■ Sketch the graphs of natural logarithmic functions.

Logarithmic Functions

■ Use properties of logarithms to simplify, expand, and condense

logarithmic expressions. ■ Use inverse properties of exponential and logarithmic functions

to solve exponential and logarithmic equations. ■ Use properties of natural logarithms to answer questions about

real-life situations.

The Natural Logarithmic Function From your previous algebra courses, you should be somewhat familiar with logarithms. For instance, the common logarithm log10 x is defined as log10 x ⫽ b

if and only if

10b ⫽ x.

The base of common logarithms is 10. In calculus, the most useful base for logarithms is the number e. Definition of the Natural Logarithmic Function

The natural logarithmic function, denoted by ln x, is defined as ln x ⫽ b

if and only if eb ⫽ x.

ln x is read as “el en of x” or as “the natural log of x.” f(x) = e x

y 3

(1, e)

y=x

2

(0, 1)

(e, 1)

(−1, e1 ) −3

−2

−1

This definition implies that the natural logarithmic function and the natural exponential function are inverse functions. So, every logarithmic equation can be written in an equivalent exponential form and every exponential equation can be written in logarithmic form. Here are some examples.

(1, 0) −1

3

4

( e1 , − 1)

−2

g(x) = f −1(x) = ln x g(x) = ln x Domain: (0, ∞) Range: (−∞, ∞) Intercept: (1, 0) Always increasing ln x → ∞ as x → ∞ ln x → −∞ as x → 0 + Continuous One-to-one

FIGURE 4.13

x

Logarithmic form:

Exponential form:

ln 1 ⫽ 0

e0 ⫽ 1

ln e ⫽ 1

e1 ⫽ e

ln

1 ⫽ ⫺1 e

ln 2 ⬇ 0.693

e⫺1 ⫽

1 e

e0.693 ⬇ 2

Because the functions f 共x兲 ⫽ e x and g共x兲 ⫽ ln x are inverse functions, their graphs are reflections of each other in the line y ⫽ x. This reflective property is illustrated in Figure 4.13. The figure also contains a summary of several properties of the graph of the natural logarithmic function. Notice that the domain of the natural logarithmic function is the set of positive real numbers—be sure you see that ln x is not defined for zero or for negative numbers. You can test this on your calculator. If you try evaluating ln共⫺1兲 or ln 0, your calculator should indicate that the value is not a real number.

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318

CHAPTER 4

Exponential and Logarithmic Functions

Example 1

Graphing Logarithmic Functions

Sketch the graph of each function. TECHNOLOGY What happens when you take the logarithm of a negative number? Some graphing utilities do not give an error message for ln共⫺1兲. Instead, the graphing utility displays a complex number. For the purpose of this text, however, it is assumed that the domain of the logarithmic function is the set of positive real numbers.

a. f 共x兲 ⫽ ln共x ⫹ 1兲

b. f 共x兲 ⫽ 2 ln共x ⫺ 2兲

SOLUTION

a. Because the natural logarithmic function is defined only for positive values, the domain of the function is x ⫹ 1 > 0, or x > ⫺1.

Domain

To sketch the graph, begin by constructing a table of values, as shown below. Then plot the points in the table and connect them with a smooth curve, as shown in Figure 4.14(a). x

⫺0.5

0

0.5

1

1.5

2

ln共x ⫹ 1兲

⫺0.693

0

0.405

0.693

0.916

1.099

b. The domain of this function is x ⫺ 2 > 0, or x > 2.

Domain

A table of values for the function is shown below, and its graph is shown in Figure 4.14(b). x

2.5

3

3.5

4

4.5

5

2 ln共x ⫺ 2兲

⫺1.386

0

0.811

1.386

1.833

2.197

y

✓CHECKPOINT 1

3

Use a graphing utility to complete the table and graph the function.

2

f 共x兲 ⫽ ln共x ⫹ 2兲 x

⫺1.5

⫺1

⫺0.5

f 共x兲

0

0.5

f(x) = ln(x + 1)

2 1

1

1

2

x

1

−1

−1

−2

−2

(a) ■

f(x) = 2 ln(x − 2)

3

1

f 共x兲 x

y

4

5

x

(b)

FIGURE 4.14

STUDY TIP How does the graph of f 共x兲 ⫽ ln共x ⫹ 1兲 relate to the graph of y ⫽ ln x? The graph of f is a translation of the graph of y ⫽ ln x one unit to the left.

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SECTION 4.4

Logarithmic Functions

319

Properties of Logarithmic Functions Recall from Section 1.4 that inverse functions have the property that f 共 f ⫺1共x兲兲 ⫽ x

and

f ⫺1共 f 共x兲兲 ⫽ x.

The properties listed below follow from the fact that the natural logarithmic function and the natural exponential function are inverse functions. Inverse Properties of Logarithms and Exponents

1. ln e x ⫽ x

Example 2

2. eln x ⫽ x

Applying Inverse Properties

Simplify each expression. a. ln e 冪2

b. eln 3x

SOLUTION

a. Because ln e x ⫽ x, it follows that ln e冪2 ⫽ 冪2. b. Because eln x ⫽ x, it follows that eln 3x ⫽ 3x.

✓CHECKPOINT 2 Simplify each expression. a. ln e 3

b. e ln共x⫹1兲



Most of the properties of exponential functions can be rewritten in terms of logarithmic functions. For instance, the property e xe y ⫽ e x⫹y states that you can multiply two exponential expressions by adding their exponents. In terms of logarithms, this property becomes ln xy ⫽ ln x ⫹ ln y. This property and two other properties of logarithms are summarized below. STUDY TIP There is no general property that can be used to rewrite ln共x ⫹ y兲. Specifically, ln共x ⫹ y兲 is not equal to ln x ⫹ ln y.

Properties of Logarithms

1. ln xy ⫽ ln x ⫹ ln y

2. ln

x ⫽ ln x ⫺ ln y y

3. ln x n ⫽ n ln x

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320

CHAPTER 4

Exponential and Logarithmic Functions

Rewriting a logarithm of a single quantity as the sum, difference, or multiple of logarithms is called expanding the logarithmic expression. The reverse procedure is called condensing a logarithmic expression. TECHNOLOGY Try using a graphing utility to verify the results of Example 3(b). That is, try graphing the functions y ⫽ ln 冪x2 ⫹ 1

Example 3

Expanding Logarithmic Expressions

Use the properties of logarithms to rewrite each expression as a sum, difference, or multiple of logarithms. (Assume x > 0 and y > 0.) a. ln

10 9

b. ln 冪x2 ⫹ 1

c. ln

xy 5

d. ln 关x2共x ⫹ 1兲兴

SOLUTION

and y⫽

a. ln 10 9 ⫽ ln 10 ⫺ ln 9

1 ln共x2 ⫹ 1兲. 2

Because these two functions are equivalent, their graphs should coincide.

Property 2

b. ln 冪x ⫹ 1 ⫽ ln共x ⫹ 1兲 ⫽ 12 ln共x2 ⫹ 1兲 2

c. ln

2

1兾2

Rewrite with rational exponent. Property 3

xy ⫽ ln共xy兲 ⫺ ln 5 5 ⫽ ln x ⫹ ln y ⫺ ln 5

Property 2 Property 1

d. ln关x2共x ⫹ 1兲兴 ⫽ ln x2 ⫹ ln共x ⫹ 1兲 ⫽ 2 ln x ⫹ ln共x ⫹ 1兲

Property 1 Property 3

✓CHECKPOINT 3 Use the properties of logarithms to rewrite each expression as a sum, difference, or multiple of logarithms. (Assume x > 0 and y > 0.) a. ln

2 5

3 x ⫹ 2 b. ln 冪

Example 4

c. ln

x 5y

d. ln x共x ⫹ 1兲2



Condensing Logarithmic Expressions

Use the properties of logarithms to rewrite each expression as the logarithm of a single quantity. (Assume x > 0 and y > 0.) a. ln x ⫹ 2 ln y b. 2 ln共x ⫹ 2兲 ⫺ 3 ln x

✓CHECKPOINT 4

SOLUTION

Use the properties of logarithms to rewrite each expression as the logarithm of a single quantity. (Assume x > 0 and y > 0.) a. 4 ln x ⫹ 3 ln y b. ln 共x ⫹ 1兲 ⫺ 2 ln 共x ⫹ 3兲



a. ln x ⫹ 2 ln y ⫽ ln x ⫹ ln y2 ⫽ ln xy2

Property 3

b. 2 ln共x ⫹ 2兲 ⫺ 3 ln x ⫽ ln共x ⫹ 2兲2 ⫺ ln x3 共x ⫹ 2兲2 ⫽ ln x3

Property 3

Property 1

Property 2

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SECTION 4.4

Logarithmic Functions

321

Solving Exponential and Logarithmic Equations The inverse properties of logarithms and exponents can be used to solve exponential and logarithmic equations, as shown in the next two examples. STUDY TIP In the examples on this page, note that the key step in solving an exponential equation is to take the log of each side, and the key step in solving a logarithmic equation is to exponentiate each side.

Example 5

Solving Exponential Equations

Solve each equation. a. e x ⫽ 5

b. 10 ⫹ e0.1t ⫽ 14

SOLUTION

a.

ex ⫽ 5 ln e x ⫽ ln 5 x ⫽ ln 5

Write original equation. Take natural log of each side. Inverse property: ln e x ⫽ x

b. 10 ⫹ e0.1t ⫽ 14 e0.1t ⫽ 4 ln e0.1t ⫽ ln 4 0.1t ⫽ ln 4 t ⫽ 10 ln 4

Write original equation. Subtract 10 from each side. Take natural log of each side. Inverse property: ln e0.1t ⫽ 0.1t Multiply each side by 10.

✓CHECKPOINT 5 Solve each equation. a. e x ⫽ 6

Example 6

b. 5 ⫹ e0.2t ⫽ 10



Solving Logarithmic Equations

Solve each equation. a. ln x ⫽ 5

b. 3 ⫹ 2 ln x ⫽ 7

SOLUTION

a. ln x ⫽ 5 eln x ⫽ e5 x ⫽ e5

Write original equation.

b. 3 ⫹ 2 ln x ⫽ 7 2 ln x ⫽ 4 ln x ⫽ 2 eln x ⫽ e2 x ⫽ e2

Write original equation.

Exponentiate each side. Inverse property: eln x ⫽ x

Subtract 3 from each side. Divide each side by 2. Exponentiate each side. Inverse property: eln x ⫽ x

✓CHECKPOINT 6 Solve each equation. a. ln x ⫽ 4

b. 4 ⫹ 5 ln x ⫽ 19



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322

CHAPTER 4

Exponential and Logarithmic Functions

Example 7

Finding Doubling Time

You deposit P dollars in an account whose annual interest rate is r, compounded continuously. How long will it take for your balance to double? SOLUTION

A⫽

The balance in the account after t years is

Pe rt.

So, the balance will have doubled when Pert ⫽ 2P. To find the “doubling time,” solve this equation for t. Pert ⫽ 2P e rt ⫽ 2 ln e rt ⫽ ln 2 rt ⫽ ln 2

Doubling Account Balances

Doubling time (in years)

t

24 22 20 18 16 14 12 10 8 6 4 2

1 t = ln 2 r

t⫽

Balance in account has doubled. Divide each side by P. Take natural log of each side. Inverse property: ln e rt ⫽ rt

1 ln 2 r

Divide each side by r.

From this result, you can see that the time it takes for the balance to double is inversely proportional to the interest rate r. The table shows the doubling times for several interest rates. Notice that the doubling time decreases as the rate increases. The relationship between doubling time and the interest rate is shown graphically in Figure 4.15. 0.04 0.08 0.12 0.16 0.20

Interest rate

r

r

3%

4%

5%

6%

7%

8%

9%

10%

11%

12%

t

23.1

17.3

13.9

11.6

9.9

8.7

7.7

6.9

6.3

5.8

FIGURE 4.15

✓CHECKPOINT 7 Use the equation found in Example 7 to determine the amount of time it would take for your balance to double at an interest rate of 8.75%. ■

CONCEPT CHECK 1. What are common logarithms and natural logarithms? 2. Write “logarithm of x with base 3” symbolically. 3. What are the domain and range of f 冇x冈 ⴝ ln x? 4. Explain the relationship between the functions f 冇x冈 ⴝ ln x and g冇x冈 ⴝ e x.

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SECTION 4.4

Skills Review 4.4

Logarithmic Functions

323

The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 0.1, 0.3, and 4.2.

In Exercises 1–8, use the properties of exponents to simplify the expression. 1. 共4 2兲共4⫺3兲

2. 共23兲 2

3.

34 3⫺2

5. e 0

6. 共3e兲 4

7.

冢e2 冣

⫺1

3

4.

冢32冣

8.

冢4e25 冣

⫺3

2 ⫺3兾2

In Exercises 9–12, solve for x. 9. 0 < x ⫹ 4

10. 0 < x2 ⫹ 1

11. 0 < 冪x2 ⫺ 1

12. 0 < x ⫺ 5

In Exercises 13 and 14, find the balance in the account after 10 years. 13. P ⫽ $1900, r ⫽ 6%, compounded continuously 14. P ⫽ $2500, r ⫽ 3%, compounded continuously

Exercises 4.4

See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–8, write the logarithmic equation as an exponential equation, or vice versa. 1. ln 2 ⫽ 0.6931 . . .

2. ln 9 ⫽ 2.1972 . . .

3. ln 0.2 ⫽ ⫺1.6094 . . .

4. ln 0.05 ⫽ ⫺2.9957 . . .

5. e0 ⫽ 1

6. e2 ⫽ 7.3891 . . .

7. e⫺3 ⫽ 0.0498 . . .

8. e0.25 ⫽ 1.2840 . . .

In Exercises 9–12, match the function with its graph. [The graphs are labeled (a)–(d).] y

(a)

y

(b)

1 2

3

−1 y

(c)

x

1

2

1 −1

1

1

2

3

x

16. y ⫽ 5 ⫹ ln x

17. y ⫽ 3 ln x

1 18. y ⫽ 4 ln x

In Exercises 19–22, analytically show that the functions are inverse functions. Then use a graphing utility to show this graphically.

g共x兲 ⫽ 12 ⫹ ln 冪x

23. ln e x

1

3

ⱍⱍ

14. y ⫽ ln x

15. y ⫽ ln 2x

20. f 共x兲 ⫽ e x ⫺ 1 g共x兲 ⫽ ln共x ⫹ 1兲 22. f 共x兲 ⫽ e x兾3 g共x兲 ⫽ ln x 3

In Exercises 23–28, apply the inverse properties of logarithmic and exponential functions to simplify the expression.

y

(d)

12. f 共x兲 ⫽ ⫺ln共x ⫺ 1兲

13. y ⫽ ln共x ⫺ 1兲

21. f 共x兲 ⫽ e2x⫺1

−1

10. f 共x兲 ⫽ ⫺ln x

In Exercises 13–18, sketch the graph of the function.

g共x兲 ⫽ ln 冪x

1

−2

11. f 共x兲 ⫽ ln共x ⫹ 2兲

19. f 共x兲 ⫽ e 2x

2 x

9. f 共x兲 ⫽ 2 ⫹ ln x

2

3

x

2

24. ln e 2x⫺1

25. e ln共5x⫹2兲

26. e ln 冪x

27. ⫺1 ⫹ ln e 2x

28. ⫺8 ⫹ e ln x

3

−2

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324

CHAPTER 4

Exponential and Logarithmic Functions

In Exercises 29 and 30, use the properties of logarithms and the fact that ln 2 y 0.6931 and ln 3 y 1.0986 to approximate the logarithm. Then use a calculator to confirm your approximation. 3 (b) ln 2

29. (a) ln 6 30. (a) ln 0.25

(b) ln 24

(d) ln 冪3

(c) ln 81 (c)

3 12 ln 冪

(d) ln

1 72

In Exercises 31–40, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. 2 31. ln 3

1 32. ln 5

33. ln xyz

xy 34. ln z

35. ln 冪x 2 ⫹ 1

36. ln

37. ln 关z共z ⫺ 1兲2兴

3 x2 ⫹ 1 38. ln 共x 冪 兲

39. ln

3x共x ⫹ 1兲 共2x ⫹ 1兲2

3

42. ln共2x ⫹ 1兲 ⫹ ln共2x ⫺ 1兲

43. 3 ln x ⫹ 2 ln y ⫺ 4 ln z

1 44. 2 ln 3 ⫺ 2 ln共x2 ⫹ 1兲

45. 3关ln x ⫹ ln共x ⫹ 3兲 ⫺ ln共x ⫹ 4兲兴

r

2%

4%

6%

8%

1 48. 2 关 ln x ⫹ 4 ln共x ⫹ 1兲兴

ln共x ⫹ 1兲 ⫺ 23 ln 共x ⫺ 1兲 ln共x ⫺ 2兲 ⫹ 32 ln共x ⫹ 2兲







51. e ln x ⫽ 4

52. e ln x ⫺ 9 ⫽ 0

53. ln x ⫽ 0

54. 2 ln x ⫽ 4

P ⫽ 131e0.019t

55. ln 2x ⫽ 2.4

56. ln 4x ⫽ 1

57. 3 ln 5x ⫽ 10

58. 2 ln 4x ⫽ 7

where t ⫽ 0 corresponds to 1980. Bureau)

2

60. e⫺0.5x ⫽ 0.075

⫽4

61. 300e⫺0.2t ⫽ 700

62. 400e⫺0.0174t ⫽ 1000

63. 4e2x⫺1 ⫺ 1 ⫽ 5

64. 2e⫺x⫹1 ⫺ 5 ⫽ 9

65.

10 ⫽ 2.5 1 ⫹ 4e⫺0.01x

66.

50 ⫽ 10.5 1 ⫹ 12e⫺0.02x

67. 52x ⫽ 15

68. 21⫺x ⫽ 6

69. 500共1.07兲t ⫽ 1000

70. 400共1.06兲t ⫽ 1300



1⫹

73.



0.878 16 ⫺ 26

0.07 12



12t

71.

⫽3



3t

⫽ 30

14%

81. Population Growth The population P (in thousands) of Orlando, Florida from 1980 through 2005 can be modeled by

In Exercises 51–74, solve for x or t.

59.

12%

79. Demand The demand function for a product is given by 4 p ⫽ 5000 1 ⫺ 4 ⫹ e⫺0.002x where p is the price per unit and x is the number of units sold. Find the numbers of units sold for prices of (a) p ⫽ $200 and (b) p ⫽ $800.



3 47. 2 关ln x共x2 ⫹ 1兲 ⫺ ln共x ⫹ 1兲兴

e x⫹1

10%

80. Demand The demand function for a product is given by 3 p ⫽ 10,000 1 ⫺ 3 ⫹ e⫺0.001x where p is the price per unit and x is the number of units sold. Find the numbers of units sold for prices of (a) p ⫽ $500 and (b) p ⫽ $1500.

1 46. 3 关2 ln共x ⫹ 3兲 ⫹ ln x ⫺ ln共x2 ⫺ 1兲兴

50.

77. Compound Interest A deposit of $1000 is made in an account that earns interest at an annual rate of 5%. How long will it take for the balance to double if the interest is compounded (a) annually, (b) monthly, (c) daily, and (d) continuously?

t

2x

41. ln共x ⫺ 2兲 ⫺ ln共x ⫹ 2兲

1 3 1 2

76. r ⫽ 0.12

冪x2 ⫺ 1

In Exercises 41–50, write the expression as the logarithm of a single quantity.

49.

75. r ⫽ 0.085

78. Compound Interest Use a spreadsheet to complete the table, which shows the time t necessary for P dollars to triple if the interest is compounded continuously at the rate of r.

冪x ⫹x 1

40. ln

In Exercises 75 and 76, $3000 is invested in an account at interest rate r, compounded continuously. Find the time required for the amount to (a) double and (b) triple.

12t

72.



1⫹

74.



2.471 4⫺ 40

0.06 12





9t

⫽5 ⫽ 21

(Source: U.S. Census

(a) According to this model, what was the population of Orlando in 2005? (b) According to this model, in what year will Orlando have a population of 300,000? 82. Population Growth The population P (in thousands) of Houston, Texas from 1980 through 2005 can be modeled by P ⫽ 1576e0.01t, where t ⫽ 0 corresponds to 1980. (Source: U.S. Census Bureau) (a) According to this model, what was the population of Houston in 2005? (b) According to this model, in what year will Houston have a population of 2,500,000?

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SECTION 4.4 Carbon Dating In Exercises 83–86, you are given the ratio of carbon atoms in a fossil. Use the information to estimate the age of the fossil. In living organic material, the ratio of radioactive carbon isotopes to the total number of carbon atoms is about 1 to 1012. (See Example 2 in Section 4.1.) When organic material dies, its radioactive carbon isotopes begin to decay, with a half-life of about 5715 years. So, the ratio R of carbon isotopes to carbon-14 atoms is modeled by t 5715 R ⴝ 10ⴚ12共12兲 / , where t is the time (in years) and t ⴝ 0 represents the time when the organic material died. 83. R ⫽ 0.32 ⫻ 10⫺12

84. R ⫽ 0.27 ⫻ 10⫺12

85. R ⫽ 0.22 ⫻ 10⫺12

86. R ⫽ 0.13 ⫻ 10⫺12

87. Learning Theory Students in a mathematics class were given an exam and then retested monthly with equivalent exams. The average scores S (on a 100-point scale) for the class can be modeled by S ⫽ 80 ⫺ 14 ln共t ⫹ 1兲, 0 ≤ t ≤ 12, where t is the time in months. (a) What was the average score on the original exam? (b) What was the average score after 4 months? (c) After how many months was the average score 46? 88. Learning Theory In a group project in learning theory, a mathematical model for the proportion P of correct responses after n trials was found to be 0.83 . 1 ⫹ e⫺0.2n (a) Use a graphing utility to graph the function. P⫽

(b) Use the graph to determine any horizontal asymptotes of the graph of the function. Interpret the meaning of the upper asymptote in the context of the problem. (c) After how many trials will 60% of the responses be correct? 89. Agriculture The yield V (in pounds per acre) for an orchard at age t (in years) is modeled by V ⫽ 7955.6e⫺0.0458兾t. (a) Use a graphing utility to graph the function. (b) Determine the horizontal asymptote of the graph of the function. Interpret its meaning in the context of the problem. (c) Find the time necessary to obtain a yield of 7900 pounds per acre. 90. MAKE A DECISION: FINANCE You are investing P dollars at an annual interest rate of r, compounded continuously, for t years, Which of the following options would you choose to get the highest value of the investment? Explain your reasoning. (a) Double the amount you invest. (b) Double your interest rate. (c) Double the number of years.

Logarithmic Functions

325

91. Demonstrate that ln x x ⫽ ln ⫽ ln x ⫺ ln y ln y y by using a spreadsheet to complete the table. ln x ln y

x

y

1

2

3

4

10

5

4

0.5

ln

x y

ln x ⫺ ln y

92. Use a spreadsheet to complete the table using f 共x兲 ⫽ x

1

5

10

10 2

10 4

ln x . x

10 6

f 共x兲 (a) Use the table to estimate the limit: lim f 共x兲. x→ ⬁

(b) Use a graphing utility to estimate the relative extrema of f. In Exercises 93 and 94, use a graphing utility to verify that the functions are equivalent for x > 0. x2 93. f 共x兲 ⫽ ln 94. f 共x兲 ⫽ ln 冪x 共x 2 ⫹ 1兲 4 g共x兲 ⫽ 2 ln x ⫺ ln 4 g共x兲 ⫽ 12关ln x ⫹ ln共x2 ⫹ 1兲兴 True or False? In Exercises 95–100, determine whether the statement is true or false given that f 冇x冈 ⴝ ln x. If it is false, explain why or give an example that shows it is false. 95. f 共0兲 ⫽ 0 96. f 共ax兲 ⫽ f 共a兲 ⫹ f 共x兲,

a > 0, x > 0

97. f 共x ⫺ 2兲 ⫽ f 共x兲 ⫺ f 共2兲,

x > 2

1 2

98. 冪f 共x兲 ⫽ f 共x兲 99. If f 共u兲 ⫽ 2 f 共v兲, then v ⫽ u2. 100. If f 共x兲 < 0, then 0 < x < 1. 101. Research Project

冢10 ⫹

y ⫽ 10 ln

Use a graphing utility to graph

冪100 ⫺ x 2

10

冣⫺

冪100 ⫺ x 2

over the interval 共0, 10兴. This graph is called a tractrix or pursuit curve. Use your school’s library, the Internet, or some other reference source to find information about a tractrix. Explain how such a curve can arise in a real-life setting.

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326

CHAPTER 4

Exponential and Logarithmic Functions

Section 4.5

Derivatives of Logarithmic Functions

■ Find derivatives of natural logarithmic functions. ■ Use calculus to analyze the graphs of functions that involve the

natural logarithmic function. ■ Use the definition of logarithms and the change-of-base formula to

evaluate logarithmic expressions involving other bases. ■ Find derivatives of exponential and logarithmic functions involving

other bases.

Derivatives of Logarithmic Functions D I S C O V E RY Sketch the graph of y ⫽ ln x on a piece of paper. Draw tangent lines to the graph at various points. How do the slopes of these tangent lines change as you move to the right? Is the slope ever equal to zero? Use the formula for the derivative of the logarithmic function to confirm your conclusions.

Implicit differentiation can be used to develop the derivative of the natural logarithmic function. y ⫽ ln x ey ⫽ x d y d 关e 兴 ⫽ 关x兴 dx dx dy ey ⫽ 1 dx dy 1 ⫽ y dx e dy 1 ⫽ dx x

Natural logarithmic function Write in exponential form. Differentiate with respect to x. Chain Rule Divide each side by e y. Substitute x for e y.

This result and its Chain Rule version are summarized below. Derivative of the Natural Logarithmic Function

Let u be a differentiable function of x. 1.

d 1 关ln x兴 ⫽ dx x

Example 1

2.

d 1 du 关ln u兴 ⫽ dx u dx

Differentiating a Logarithmic Function

Find the derivative of f 共x兲 ⫽ ln 2x. SOLUTION

shown. f⬘共x兲 ⫽

Let u ⫽ 2x. Then du兾dx ⫽ 2, and you can apply the Chain Rule as 1 du 1 1 ⫽ 共2兲 ⫽ u dx 2x x

✓CHECKPOINT 1 Find the derivative of f 共x兲 ⫽ ln 5x.



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SECTION 4.5

Example 2

Derivatives of Logarithmic Functions

Differentiating Logarithmic Functions

Find the derivative of each function. a. f 共x兲 ⫽ ln共2x 2 ⫹ 4兲 STUDY TIP When you are differentiating logarithmic functions, it is often helpful to use the properties of logarithms to rewrite the function before differentiating. To see the advantage of rewriting before differentiating, try using the Chain Rule to differentiate f 共x兲 ⫽ ln冪x ⫹ 1 and compare your work with that shown in Example 3.

b. f 共x兲 ⫽ x ln x

c. f 共x兲 ⫽

ln x x

SOLUTION

a. Let u ⫽ 2x 2 ⫹ 4. Then du兾dx ⫽ 4x, and you can apply the Chain Rule. f⬘共x兲 ⫽ ⫽ ⫽

1 du u dx 2x 2 x2

Chain Rule

1 共4x兲 ⫹4

2x ⫹2

Simplify.

b. Using the Product Rule, you can find the derivative. d d 关ln x兴 ⫹ 共ln x兲 关x兴 dx dx 1 ⫽x ⫹ 共ln x兲共1兲 x ⫽ 1 ⫹ ln x

f⬘共x兲 ⫽ x

Product Rule

冢冣

✓CHECKPOINT 2 Find the derivative of each function. a. f 共x兲 ⫽ ln共x 2 ⫺ 4兲 b. f 共x兲 ⫽

x2

ln x

c. f 共x兲 ⫽ ⫺

ln x x2

Simplify.

c. Using the Quotient Rule, you can find the derivative. d d 关ln x兴 ⫺ 共ln x兲 关x兴 dx dx f⬘共x兲 ⫽ x2 1 x ⫺ ln x x ⫽ x2 x

Quotient Rule

冢冣





1 ⫺ ln x x2

Example 3

Simplify.

Rewriting Before Differentiating

Find the derivative of f 共x兲 ⫽ ln冪x ⫹ 1. SOLUTION

f 共x兲 ⫽ ln冪x ⫹ 1 ⫽ ln共x ⫹ 1兲 1兾2

✓CHECKPOINT 3 Find the derivative of 3 x ⫹ 1. ■ f 共x兲 ⫽ ln 冪

1 ⫽ ln共x ⫹ 1兲 2 1 1 f⬘共x兲 ⫽ 2 x⫹1





1 2共x ⫹ 1兲



Write original function. Rewrite with rational exponent. Property of logarithms Differentiate. Simplify.

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327

328

CHAPTER 4

Exponential and Logarithmic Functions

D I S C O V E RY What is the domain of the function f 共x兲 ⫽ ln冪x ⫹ 1 in Example 3? What is the domain of the function f ⬘ 共x兲 ⫽ 1兾关2共x ⫹ 1兲兴? In general, you must be careful to understand the domains of functions involving logarithms. For example, are the domains of the functions y1 ⫽ ln x 2 and y2 ⫽ 2 ln x the same? Try graphing them on your graphing utility. The next example is an even more dramatic illustration of the benefit of rewriting a function before differentiating.

Example 4

Rewriting Before Differentiating

Find the derivative of f 共x兲 ⫽ ln 关x共x 2 ⫹ 1兲 2兴 . SOLUTION

f 共x兲 ⫽ ln 关x共x 2 ⫹ 1兲2兴 ⫽ ln x ⫹ ln共x 2 ⫹ 1兲2 ⫽ ln x ⫹ 2 ln共x 2 ⫹ 1兲 2x 1 f⬘共x兲 ⫽ ⫹ 2 2 x x ⫹1







1 4x ⫹ 2 x x ⫹1

Write original function. Logarithmic properties Logarithmic properties Differentiate. Simplify.

✓CHECKPOINT 4 Find the derivative of f 共x兲 ⫽ ln 关x2冪x2 ⫹ 1 兴.



STUDY TIP Finding the derivative of the function in Example 4 without first rewriting would be a formidable task. f⬘共x兲 ⫽

x共

x2

1 d 关x共x 2 ⫹ 1兲2兴 2 ⫹ 1兲 dx

You might try showing that this yields the same result obtained in Example 4, but be careful—the algebra is messy.

TECHNOLOGY A symbolic differentiation utility will not generally list the derivative of the logarithmic function in the form obtained in Example 4. Use a symbolic differentiation utility to find the derivative of the function in Example 4. Show that the two forms are equivalent by rewriting the answer obtained in Example 4.

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SECTION 4.5

Derivatives of Logarithmic Functions

329

Applica